I believe that to open a normal "onshore" UK bank account then you need to have a UK address - at least this is in the terms and conditions of any banks that I have tried to open an account with.
If you do come across this hurdle then an option maybe one of the "offshore" banks in the Isle of Man - there are several that you could find through google - but you will find that fees are higher than if you manage to open one in "the North" as TheShark suggested.
My own situation was that I was receiving a monthly £Stg payment by cheque which was a pain. I called First Trust (AIB) in Newry ,told them what I wanted, they gave me an appointment , told me what documents I needed (usual stuff , Passport , proof of address etc.) so I had the account , complete with ATM/Debit Card up and running in a few days. Quite straightforward.
My local AIB branch set up a sterling account several years ago for me with one of their London branches ( 11 Old Jewry, London EC2 8DP ). I used to make sterling lodgements through my local AIB bank and wrote sterling cheques for payments out of the account - never had any bother.
I have the exact same set up as above. Local aIB set up a sterling bank account that is a back to back account with my euro account so if I don't have the sterling to match a payment then they will convert euro from my sterling account.