I know a guy who does quite a lot of things around the area. Haven't used him myself but he is Gary Roberts on 07833 536346. He frequently goes to Ikea to rescue people who have bought things which won't fit into their cars !!
I don't want to rain on your parade clarissajazz, but it would be prudent to check that whoever does the job has Hire and Reward insurance.
If they don't, and you and/or your stuff get damaged in an accident then you'll get zilch.
I used Rocket Van & Box Shop on Union Street and would recommend them/use them again. I moved over Easter weekend 2011 from The Cut to Bermondsey. I cannot remember exactly but the move took between 2 -2.5 hours with two guys working and the cost was reasonable and inline with the estimate. The driver even gave me a ride back to my old flat on his way to the next job which was nice.
The shop is under the arches near Southwark Station. They delivered the packing supplies to my flat in advance of the move. I had been a customer of the Box Shop previously - they sell sturdy plastic storage containers with lids that actually stay on!