Moved last week within Basel and can highly recommend Top Speedy Removals, as recommended to me.
http://www.tsr-removals.ch

They were the most competitive by far, the quote was on the nose, Herr Kilim was on the ball, they were fast and coped with all the miscellaneous bric-a-brac I'd failed to deal with.
Two vans and five men meant I was in the new place unpacking before midday.

Also plugged here (but with poor tags)
Moving Company in Basel

and they ate all the cheese sarnies I supplied.

but you denied us the fun of getting together so that a whole bunch of Basel EFers could help you move everything, and then eat all your sandwiches. :-)

Hi. Do you have any idea if they would move my stuff back to the UK?

Looked to be Swiss orientated to me.

We used Team Allied to move from Luxembourg to Switzerland, they have a slightly different mode of operation, in that they use local crews to do packing and loading/unloading and only send the driver with the truck, cuts down on expenses so makes them a little cheaper.