For anyone thinking about moving to the UK (or maybe any European country), here's some tips that I wish I had when I came over.
The banks that have been the most helpful:
NatWest required proof of address, and suggested that I get my bank at home to mail a statement to my temporary address here in the UK. Had I known that earlier, this would have been the best option.
HSBC has an account for incoming international people. You can (and should) apply for it before you come to the UK. They call it the Passport account. They do some background checks that took about a week for me. They then required me to prove my address back in the States. Unlike other banks, they accepted my proof of employment letters as proof of address (they had my US address on them.) The one down side is that their account costs 8 pounds per month. When you can't get an account any other way, that doesn't sound like so much.
The bank that sort of considered helping me, but screwed up with paperwork and cost me 3 weeks:
The co-operative bank
The banks that outright denied me any consideration:
Royal Bank of Scotland