If you have an employer, they can write a letter for you verifying your address. This is what I did (this letter + passport) and opened an account at Barclays.
First tried Natwest with similar letter, but they wanted 6 months of bank statements from my overseas bank plus set up an appointment first (and "none available today"). The guy was so haughty I told him I'll just go elsewhere when asked if I wanted to make an appointment. So frustrating.
I had an easier time opening bank accounts in the US (seems to require only a wad of cash
and Australia (passport with visa enough), but things may have changed there recently. On the up side, in the UK, banking is free. No teller fees, no lack of activity fees, you can use other banks' ATMs for free, and my required minimum balance with Barclays is zero.
To those still in their home countries, save your bank statements just in case!