We used James Removals - not too expensive and we didn't have to buy the boxes as they reimburse you when you give them back (they pick them up and don't care if a few are missing due to them being ripped or whatever). We packed ourselves although they do have a packing service (but I am weird about people touching my stuff!) and they managed to move us in under 6 hours (it was only up the road but I have seen companies take 8 hours just to pack up never mind unload). Lovely friendly company.
I absolutely and highly recommend the excellent Jan and team at JT Enterprises based in East Dulwich. Totally fab guy and so hardworking. His team were amazing and we were very last minute due to very difficult chain and exchange/completion happening within hours of each other! His no is 07850 716079.