They're similar to Zipcar/Zipvan and the other car clubs, but with no cars around here; only vans! You can hire them by the hour with no deposit and no exorbitant membership fee - I think it was £3 to join and that covers the cost of them conferencing calling the DVLA to check.
We got a Transit from them for £42 for the day (24 hours) and it's a self service with no deposit. That was for a weekday; I think it's more at the weekend, but not by too much.
Alternatively, try the London Transit Van Centre on the Old Kent Road. They're very good and offer excellent customer service, but offer more traditional van hire by the day, week and weekend.
I used Rocket Van http://www.rocketvan.co.uk/ for my local move in 2011 for a 2 bd flat and highly recommend them. They were fast, careful and seemed good value. They are on Union Street near Southwark Station. It was cheaper to hire Rocket for 2 hours (2 men + truck) than it was to use ZipCar in a previous move. I used ZipCar/Van in 2010 and it ended up taking 6 hours to move as we had to take several trips between the flats and we were not as fast loading and unloading. If you include the membership fee as we never used the service again, the move was twice as much as I estimated it to be.