Anyone with limited proof of ID, can open a basic bank account
Those without ID (passport, utility bills etc) can still apply for a 'Basic' bank account, which usually just provides a debit card and online banking facilities. The Benefits Agency and local authorities also strongly suggest to claimants to open a 'Basic' account. My banks gives leaflets suggesting they will open accounts from people who just provide letters from GP's/Teachers/Social Workers, confirming they know the holder, other things like Benefits Agency letters, NHS registration cards, Electoral registers all support opening such basic bank accounts.
Check out:
http://www.google.co.uk/search?hl=en&source=hp&q=basic+bank+accounts&btnG=Google+Search&meta=&aq=f&oq=