Hi Doug,
If your not too bothered about the actual phone features and just need to make phone calls then it would probably be cheaper to to buy a Pay As You Go (PAYG) mobile phone along with the sim card. A basic Nokia 6210 mobile can be picked up from an Orange shop or other retailers such as Phone4U. These retailers will be in every city centre high street. Even the local supermarkets such as Tesco (Tesco are everywhere throughout the UK) usually have a good deals on basic handsets in there larger stores.
http://www.phones4u.co.uk/shop/shop_payg_details.asp?ItemKey=156574&NetKey=4http://direct.tesco.com/q/R.202-3677.aspxA handset such as the Nokia 6210 will cost not much more than just the SIM card on its own.
Expect to pay around $20-25 for the basic PAYG handset which will include the SIM Card and around another $20 of call credit to get you going.
You will probably spend more on transport arranging to meet up with someone to lend you a handset, as any form of transport is very expensive, especially in London.