rocketvan, based in Union Street SE1. website doesn't seem to be fully up and running yet but has contact info on it. Had a (very reasonable) quote from them recently, but haven't used them yet because of change of plans.
Where are you moving to, Button? Reason I'm asking is that removal costs do vary, depending on area. If you're staying in the southeast then they're probably much of a muchness, but if your relocating to what may be a less expensive area, then it's worth getting a quote from a firm where you're moving to. I did this some years ago and saved my mother-in-law £300.
Yes, out of London companies have out of London fixed costs, wages, insurance, offices... but make sure you give them proper directions, and make proper provision for their loading/parking spaces. They may struggle with stairs, and lifts, and getting furniture through doorways tho'.