Where / how can I purchase a cell phone in UK ?

Bill, Nigel provided an excellent description of the best places to look for a Cell phone (usually referred to as a "Mobile" in the U.K.). The U.K. is one of the easiest and cheapest places in Europe to buy these, and you'll find that Cell shops are ubiquitous in most cities. Both network stores (O2, Orange, Vodaphone, T-Mobile, "3") and resellers such as Carphone Warehouse, Tesco, Virgin or others are well represented. As Nigel mentioned, firms such as Tesco and Virgin are classed as "Mobile Virtual Network Operators" (MVNO's) in that they don't actually operate a cellular network, but rather buy time from established networks and then resell under their own brand. When I transited through Heathrow last year, I noticed there are now vending machines in the airport that sell SIM cards. If you had an unlocked quad-band phone, you wouldn't even have to visit a shop (although that would be a better idea, so that you can have a choice of carriers and plans). Buying a phone in the U.K. will provide the cheapest rates and will give you a local number, should anyone from there need to call you. Most networks offer a variety of prepaid (PAYG) plans, so be sure to have a look at which one will fit your situation most appropriately. As I recall, phones in the U.K. can be topped-up at ATM's or by purchasing a card which are available at Kiosks, rail stations or wherever. Happy travels!

Any shopping centre, any high street, any supermarket. On most shopping streets you will pass a store selling mobiles (they're not called cell phones here) every few metres. You can pay with cash or with a credit or debit card. The three main carriers are O2 Vodafone EE (Everything Everywhere, formerly T-Mobile and Orange)(be prepared to be frustrated with their website) A much smaller player specializing in data is 3. Just the digit, 3. Supermarkets like Tesco run their own networks by utilizing bandwidth and cells provided by one of the big 3 (not the company "3"). There are also loads of independents who resell big 3 plus "3" phones. There's also Carphone Warehouse who sell all the networks plus one of their own. You are looking for "pay as you go". I'd probably go to a supermarket or Carphone Warehouse in your situation. You can google all of the above companies. I don't know as I never go to an airport these days, but I'd be surprised if there aren't phone stores at the airports. You might not get as good a deal there though as you would on the High Street.

I'd recommend Carphone Warehouse. In my experience they not only will sell you a very reasonably-priced unlocked phone + prepaid SIM card, they will take a few minutes and get it set up and working for you. I bought an unlocked phone + prepaid SIM package from a CW store in 2009 for less than US$20. Used my hotel's address as my street address on the form I filled out. I brought the phone back to London in 2010 and of course by then the minutes/account on the original SIm card were expired. But it wasn't hard (and it was inexpensive) to get a new SIM card loaded with prepaid minutes (the 20-something son of the B&B proprietor where I stayed helped me out). I topped it up with more minutes at the check-out counter at a local drugstore - unbelievably quick and easy. I still have the unlocked phone, and now also a plug adapter so I can charge it up before I leave home on my next trip to foreign parts.

Vodaphone has a store outside security in Terminal 5 at Heathrow.

Be careful about the SIMS sold from the machines at airports. There have been complaints that some do not include call time and are therefore an expensive option. You can get "free" SIMS from places like Carphone Warehouse (CW), the price you pay is for the actual call time on the SIM. IME it is better to go to a specialist mobile phone shop because they will be able advise you on the best PAYG plan for your needs. The CW site Nigel gave you has comparision tables for SIMS, including international options.

Wow. Thanks to all of you for the great information !

I wonder what make & model phone has done well for you ? (Eg., Sony, Samsung, Nokia, HTC.)