Which London SIM card vendors, shops, kiosks to buy from?

Tesco (with a phone shop)
Carphone Warehouse

Both have reasonably competent people behind the counter (or much more likely in front of the counter) and sell sims for all 4 networks and the several piggyback networks.

Is your phone compatible with all 4 frequencies?

Both Tesco and Carphone Warehouse have pretty decent websites which you can study before you arrive.

If you decide after checking their information you can also get sims from the 4 actual networks directly - they all have many many stores - EE, 3, Vodafone, and 02.

If you are in London the whole time all should work well for you - they all have individual dead zones, and in the countryside each will have other areas of less coverage. If you stick to motorways they should all work reasonably well - anywhere but Dorset, Devon or Cornwall. Rumour has it the prefer the 19th century and put up special defences against dark arts on the border - especially Cornwall and Devon.

I'm assuming you have a GSM capable phone.

Last fall, I chose O2. I was staying close to Kensington High Street so the shop was convenient and I liked the terms of the plan. For around 20 quid, I received 300 minutes and 2.5GB of data. It fit my needs perfectly. I was in and out of the store in about 5 minutes and my phone worked straight away.

Good luck...