There probably is not time for this on this particular trip but I purchased SIM cards on eBay before a trip to Italy last year. They were much less expensive then getting them through official outlets. As Am_Fear_Liath_Mor said, getting a refill card was easy at most Supermarkets. Should be even easier in the U.K. since I don't speak very effective Italian.

Also, cell phones in UK and most of Europe (as far as I know) only charge for outgoing calls. So if you don't originate many calls, you won't need to recharge much if at all. Though people calling you from the US will pay more than if calling a UK landline.