@ Cheers - When you buy a car the current owner should have the registration document (V5). They send it off to the DVLA and give you a tear-off section from it called the new keeper's slip or something like that. The DVLA will send you a new V5.
When I've bought or sold vehicles in the past, often the other party has insisted on watching me mail the V5. It's not a bad idea, as it ensures the buyer will be recognised as the keeper, and the seller won't get the new keeper's parking and speeding fines.
I've also always insisted on a detailed, written receipt for a private sale. I think technically having your name on the V5 confirms you as the registered keeper, but not necessarily the legal owner.