Recommend a simple, small cheap laptop < 15" I can chuck in my bag for use in coffee shops!

Refurbished Thinkpad. The answer is always refurbished Thinkpad.
- Yep. I'm using a used ThinkPad X1 Carbon. 8 years old and running Linux like a dream
  
  what are they like for duability - e.g. knocks from being put in and pulled out of a rucksack
- Lenovo also sells older models for dirt cheap on their website sometimes.

Instead of just throwing random preferences out there, I'll help clarify the field of comments:
Thinkpads USED to be a safe choice, but Lenovo has been tainting that model line for a few years. Search and find specific models, and don't just buy because it has the Thinkpad brand.
Framework is 100% ready to go. They have a Refurb store where everything is cheap, but if you find one cheaper, get it.
Dell had a ton of Linux ready laptops under the XPS brand not long ago. Search and find out which to make sure, but they shipped with Linux installed.
I hate to say it, but HP Probooks were solid and shipped with Linux also. Terrible company, but they make decent enterprise products. They'll sell for cheap on eBay.
- There are <250USD used frameworks?
  
  No, I don't think I ever seem one for under 700 USD, despite some "news" saying you could find at 500 USD.
  Every time people ask for cheap computers, there is always people sharing their preferences without any regard for OP's listed needs.
- Thinkpad T, P, W, & X (Carbon) are generally pretty solid, though T & X probably better fit OP’s preference for portability. The T series is/was also user upgradable (memory and SSD), usually pretty easily. I think some of the carbon models were also upgradable, but can’t remember. Cruicial’s website is very helpful with this. If the laptop has “Idea” or “Yoga” in the name, it’s more than likely trash. There were some “higher end” Yoga models, but AFAIR none were upgradable.
  
  old T (until t440p) you could upgrade the cpu as well, and they are dirt cheap on aliexpress
  
  Lookup the T38
- I have had more problems with two different Frameworks than most Thinkpads. Battery died, boot/power problems on both the 13 and 16, touchpad problem on the 13.
  I prefer the concept of the Frameworks but can’t say they have worked better in practice.
- https://www.dellrefurbished.com/
  Check support but can score some great deals on business-class laptops here. They’re machines that are coming off lease from Dell Financial Services.
- Great advice. Framework is the best choice if you can afford it. Seconded your opinions on Lenovo. They're absolute trash now.
- Framework is a US company and nobody wants to pay a premium to advance fascism thank you very much.
  Thinkpads are a safe choice. I have the same use case as OP and i use one. Battery last 7-8h of light use, plenty for a plugless day's work.

People are going to say Thinkpad but I am going to say 2013 to 2017 MacBook Air. Inexpensive. Light. Good looking. A joy to use. Faster than you think. And well supported under Linux (I use EndeavourOS on them myself).
- This was my immediate thought. An M1 Mac laptop is still a very useable laptop, and the battery life on them is fantastic.
  
  M1 mac is still problematic with Linux, with only Asahi offering limited support, and you cannot find one at OP's price point.
  
  A problem with macbooks is that most have soldered ram, and even hard disks, which negates the possibility to upgrade to modern capacities. Many older non apple laptops can be upgraded to a larger SSD and ram, and be given a new lease on life.
  
  M1 works decently with Linux these days but anything newer than that barely works.
  I was thinking Intel era given the price-point.
- I second this if he can find a 2013~2017 for free/cheaper, I use an old Macbook Air with EndeavourOS mostly for writing, but my use case is not that far from OP's, it also runs some old games.
  It is light and portable, and it is just as easy to find a refurbished option as the Thinkpad (at least in NA).
  My only problems are with the drivers(the Wi-FI and the FacetimeHD camera) and the charging cable.
  I tried a bunch of distros, but only with EndeavourOS I got them to work easily.
  That said, if OP prefers better Linux support, and better cost x performance. He will be better suited with a Thinkpad T or X (T480/T490 or X280) refurbished will be in the price range.
  X280 is barely above 1Kg, with a smaller screen.
  T480 is chunkier, bigger screen and Ethernet port.
  https://www.hotukdeals.com/deals/lenovo-thinkpad-x280-core-i7-8550u-16gb-ram-256gb-touchscreen-windows-11-laptop-12-months-guarantee-4640028
  https://www.ebay.co.uk/itm/184872920970

The generic answer for this is to get a refurbished thinkpad. Pretty much any T-series fit your needs and there's plenty of pre-leased corporate machines around which are refurbished and often have even a some kind of warranty.
- This...is not the best answer. You need to be REALLY SPECIFIC about model numbers now that Lenovo has pollutes that brand space.
  
  polluted, how?
  
  While you’re generally right, the T-series is a solid business laptop. Only thing I would add is steer clear of anything with “Yoga” in the name. They can be sleek, but very few of those ever impressed me.

Used Thinkpad X or T series
- What's the main difference between the two?
  
  X series are lighter and smaller than T series, on the other hand they are less upgradable.
  
  X is lighter but have already soldered in RAM. T series are a little bit more bulky but pretty much anything can be customized. Be wary of the t480 and t490 though. Those have some flimsy charging ports that if broken will be impossible to fix.

I have recently bought 2 dirt cheap thinkpads, one for me and one for the wife.
T490s - i5 intel
T14s - Ryzen 5 AMD
Both are tick all your requirements except for the numbpad, the T14s is definitely worth the extra money, though. It can even handle some medium gaming.
Both have upgradeable nvme ssds. However the ram is soldered on the "s" versions of these laptops so find one with 16gb or more.

loving my recently acquired T480. not the fastest by any means, but solid and great keyboard. plus non-soldered memory allows for upgrades. got mine off craigslist for 120, a steal.
- T480s is my backup work laptop. Runs Linux fine (have had Ubuntu, currently Fedora 42). Runs windows 11 like shit, but then my primary P1 gen 4 also doesn’t run 11 much better, so…
  
  You basically need a supercomputer for windows 11 to not run like complete shit. Linux will run well on 15 year old hardware, although I wouldn't suggest anything that old if you care about power consumption.
- Thanks for mentioning the actual model number.

A refurbished Thinkpad T480.

If you're able to code from a terminal, and care about longer battery life (my main concern when working from a coffee shop or elsewhere), I'd recommend getting a used android tablet, pry something from xiaomi or oneplus. You can find a decent model used for around that price with > 8 hour battery life easily.
Get a good stand, a solid bluetooth keyboard (logitech makes some great portable ones), and put termux on it (can probably handle light python locally).
If you need it to do CPU powerful tasks, use termux to remote into a VPS or your home server, and let a plugged in linux machine do the work so you can save your device's battery life. This is how I code at least.
- So buy a shell of a device to then SSH into a computer I leave running 24/7? Got it. Forgot to say, I pay my own bills.
  
  You mean the power cost of a computer idling at home?
  I have 5 computers (beelink and nuc servers) at home rn, each idles at ~6 watts. That's about 40 usd a year. One computer would be 8.5 usd per year.

If you're in the US, refurbished thinkpads are probably the best option. Not so much here in Australia (but you mentioned GBP so perhaps you're in UK). Whatever. I bought a refurb Dell Latitude 3120 for AU$229
mfg yr 2021 Intel(R) Pentium(R) Silver N6000 @ 1.10GHz 8Gb RAM Intel UHD graphics Intel Wi-Fi 5 9560 (160 MHz) Bluetooth 5.0 Display: 1366 x 768 11.6" touchscreen 2-in-1 Disc: M.2 256Gb PCIe NVME Class 35 SSD 1.35kg
Runs voidlinux like it was born to it. It's my travel laptop.

Dell latitude 14 inch 5430 or similar, cheap ish. Its got all the wants and needs. Plenty of ports. Its dell so it'll survive forever.

Keep browsing eBay and HotUKDeals. You can easily find something really good for <£200.
This 11th Gen Intel Dell laptop was going to £150 this morning and I was on the verge of ordering it and adding more RAM.
This sub is obsessed with Thinkpads, but when you're looking for a secondhand bargain then you pick up what comes opportunistically. The market is flooded with Dells because businesses change machines long before they go out of date and generally keep them in very very good condition.

t480 or t470

I've had great results with various refurbished Dell Latitudes from eBay over the years. I have a stack of about 5 or 6 of 'em and they've all run many mainstream Linux distros with fantastic out-of-the-box support. I pass 'em out to members of the household whenever a laptop is needed and they'll usually get the job done.
I'd just type in "Dell Latitude" on eBay and filter by price and such. I suspect any model with an i5 and 8GB RAM oughta be fine for light programming work. I've found sellers with high ratings (like 97% or higher) and thousands of sales are pretty reliable (and tend to have return policies in case you get a lemon). Just test all the hardware (webcam, microphone, headphone jack, USB ports, ethernet, etc) as soon as you get it.
I've saved a lot of money over the years buying secondhand, and these machines have been running without a hiccup for years of casual use.
- They're not light though!
  
  My 5290 definately is. But it's slightly thick. 12-13 inchers are still portable in my opinion.

A light Chromebook?

A refurb!

maybe search for system76 too. I found one on my local craigslist.

Thinkpad Thinkpad T480. Can find many within your price range. Great quality, most of it is replaceable including keyboard and display. Aside from actual hardware failure or damage, the one thing that gets hit long term with laptops, the battery, is super easy to replace and there is both an internal and external if you want really long life

I'm not sure how common they are outside Japan, but I have a little (about 12" I think) Panasonic "Let's Note" that I use quite a lot as a lightweight coding (and retro/indie gaming :D) device that I can throw in even my smallest bag when there's a chance I'll have to kill more than a few minutes. They're designed to be a little bit rugged. I had Ubuntu on it previously, now Mint, and the only problem I've had is that Linux somehow sees two screen brightness systems, and by default it connects the screen brightness keys to the wrong (i.e. nonexistent) one. Once I traced the problem it was a quick and painless fix.
They seem to be sold worldwide, so you may be able to get one cheaply second-hand. One thing to be careful about is the fact that in order to keep the physical size down, the RAM is soldered to the board. Mine is an older model (5th gen iCore), and has 4GB soldered on but also one SODIMM slot, so I was able to upgrade to 12GB total. But I've noticed that on most later models they got rid of the RAM slots entirely, so whatever RAM it comes with is what you're stuck with.

intel macbook air works good for me, with debian and xfce
- -1 for intel macbooks. Horrible cooling and poor hardware support. Source, I own one.
  
  Horrible cooling is the last couple of years of Intel MacBooks. Nothing to do with Linux.
  If you run them with a “balanced profil” in Linux, they run a bit slower but the fan stays quiet. It is probably what macOS does.
  Before 2019 or so, they run awesome. I mean, the newer ones are faster so they run great too even in balanced. I guess it depends what you pay for them.
  The older ones are crazy cheap these days and, in my view, great value.

Under 140, latitude 5290 (from 2018, 8350u). I bought it ~2y ago. Small, 12 inch, decent keyboard, very upgradable, decent build quality, but the LCD is horrendous. Edit: used, could probably be found under 100€.

Try Lenovo IdeaPad Slim 5.

Used DELL 5310. Intel 10th-gen, 60Whr battery (goes 8+ working hours on a charge) often 16GB RAM and at least a 256GB SSD at that price range. Upgradeable (DDR4, NVMe) too.

small tuxedo computer

I got myself an old EEE PC for exactly that purpose. (Except, substitute python with lua).
8h battery life, cost me €20 and does what it's supposed to. Just make sure you get one with an Atom N280 or better. The popular N270 is 32bit only, and more and more programs are dropping 32bit support. Some of them you can DIY compile for 32bit, some you really don't want to.
(For example, compiling Node on an Atom N270 takes around 3 days.)
I had one with an N270 first and replaced it with one with an N450 to get 64bit.
Maxed it out with 2GB RAM, a cheapo €10 SSD that maxes out SATA and overclocked it to 2GHz.
It's not fast by any stretch of the imagination, but it's totally ok for editing text files with Kate and compiling with platformio.

Considering your budget of 200 GBP / 250USD, I would recommend laptops meant for school. There are plenty of refurbished laptops out there with a decent battery condition and overall state for sale around €100. Most of these machines aren't more powerful than most entry level Chromebooks and often have a Pentium or Celeron CPU, but that's a tradeoff you'll have to make. Another advantage is that they usually come with a touch screen and decent display, which is nice if you're out and about.
- I would recommend laptops meant for school.
  This is a funny model number.

Don’t know where you are from, but I got a MacBook Pro 2015 for 150 and it can be forced to the latest macOS or just any distro of your liking. I will say that it can get quite hot and it’s recommended to switch the cooling paste and clean the fan.
For a cheap device it works fine and in this price bracket a better screen is barely possible.
I do have a surface book 2 that I will sell between 150 and 200, but I think Linux support is finicky at best. I ran popos on it, but there are just a bunch of things which would work a lot better on other devices
- As much as I want to agree with this (I have a 2018 MacBook Pro that is running t2linux), this is a horrible suggestion.
  Sure, if that's the only computer (or laptop) you already have, go for it, but Linux on Mac, at least via the t2linux project is currently shaky at best. It does work, but absolutely not as a daily driver in my opinion.
  Suspend is completely broken, the touchpad is barely usable, performance is horrible, audio quality is horrible, Bluetooth is unusable, battery life is abysmal.
  And that's not even mentioning the challenges you face installing it on your MacBook; firmware hacks, keyboard not working, etc.
  DO NOT buy a MacBook specifically to run Linux on it if it's going to be your daily driver. You will have a horrible time. Buy something more suited like a thinkpad.
  
  I have six MacBook Pro / MacBook Air computers. They all run flawlessly. Everything works. Everything, including all the social keys (screen, volume, etc). I do not have any of the problems described.
  I daily drive more than one of them.
  The best OS, in my view, for these machines is EndeavourOS, especially if you have one new enough to have a T2 chip. That said, I also have a 2020 MacBook Air running EndeavourOS. Absolutely everything works and it is quite fast but the fan will get quite loud (unlike any of my other machines). I have been meaning to replace the thermal paste in the hopes that it helps. I tend to use the older ones more as they do not have this issue.
  My recommendation would be to go for machines before 2020. The MacBook Air 2013 - 2017 is the absolute best bang for buck.
  I picked up a 2013 for $50 two years ago for a backpacking trip hoping I could use it for email and not caring if it was lost or stolen. I used it for 4 hours yesterday including for a Microsoft Teams meeting.
  
  Sucks to hear your experience being so terrible. Either I’m so coloured by the hellish experience of booting Linux on a surface (which is easy, but so many small issues after each update) or booting Linux on a mac is really easy.
  BUT
  I do agree, get something like a system76 for ease of use. I have dabbled with Linux, custom roms and other “hacky” stuff my whole life. I’m so used to it being hard that the experience these days feels as easy as booting windows

Do they sell ASUS in your area? I bought a 2nd hand 14" Vivobook with a Ryzen 5 GPU and it runs Fedora 42 like a champ. First laptop I've ever owned that I don't worry about overheating. It's been my chuck-it-in-a-backpack travel machine for two years with no problems.

I'm in the UK, and I've had decent laptops from rapidIT and ITZoo, both .co.uk as far as I remember. I can't remember which laptop I got from Rapid as it was a few years ago, but the first one had a faulty motherboard, and they swapped the laptop with no issues.
I had a Stonebook branded Clevo N750BU from ITZoo, which is an i7 based laptop, and Mint works perfectly on it. It's quite old now, so you'll probably get a newer one for your budget, but with a RAM upgrade it runs everything I throw at it. I use it for designing and editing logos in Inkscape and Krita, and for editing a website in html, css, and javascript, so anything newer should do it even better :)
- I use it for designing and editing logos in Inkscape and Krita, and for editing a website in html, css, and javascript,
  Computers from 30 years ago did that too.

Thinkpad x1 carbon gen 6, or if you're willing to up your budget a bit, a x13 is also a great fit.

If you're going to hang out in coffeeshops just get whatever has a clearly visible Apple logo and spend the rest of your money on a beanie, airpods and fake glasses.

A Chromebook?
- I really want to avoid the whole google thing.
  
  You can rip out ChromeOS and install Linux, good if you've already got an old EoL Chromebook but I'm not sure if it's worth actually buying one for it
  
  If you're buying used you're not directly funding Google.
- ....no

Not sure if it's available where you're located, but did find this listing. https://a.co/d/bztqux3
- There's also multiple T14's with the 8 core AMD variant at the moment on ebay. I recently purchased a similar version myself.
  https://www.ebay.com/itm/187345973525 - Grade C - $165
  https://www.ebay.com/itm/236180535889 - $180
  https://www.ebay.com/itm/236134873233 - $205
  https://www.ebay.com/itm/187365382026 - $253
  https://www.ebay.com/itm/157131848816 - $260
  I have both a 2014 MacBook Pro and the listed device. The T14 has a trash tier trackpad and display when compared to a MacBook, but if I was buying it to do any type of programming, I'd choose the T14 everytime if I'm comparing it to a MacBook with 8 GB of ram and a dual core processor. (when compared to a T14 that has the 8 core AMD 4750U)

I'm loving the new Snapdragon laptops, especially if you don't have any heavy (read: gaming) workloads!
- sub 200 GBP / 250USD I guess
  Last time I checked most were starting at 700+
  
  Whoops! I did miss that part, my bad!!

Just popped Arch on a Thinkpad Carbon X1 gen8. There about 300 on eBay. Great decision
I use Arch btw
- I use Arch btw
  Is that why you haven't read the op and just posted what you wanted to see written? How do your farts smell?