We were (unfortunately) moving house every 12 months or so for a while, and came across Frog Removals who were great. Small company but great service. Used them a few times. You might have trouble booking in so close to the date but they might be able to recommend someone else.
They all charge by time, so the more you can do yourself and make it easy for them, the quicker it is. We usually pack all our own boxes, dismantle furniture, and have it all waiting as close to the place they get the truck parked as possible.
Put everything you possibly can into a box. Like, don’t leave all the loose crap in shopping bags. We’d move flat screen tvs, computers and clothes ourselves too to make it easy and quick.
For what it's worth I'd avoid the super cheap ones.
I had a bad experience with JB Removals whose truck door didn't look like it closed properly and they damaged my fridge.
The next move we paid more for Black Stallion and they did an amazing job.