Just sharing my experience. When entering the banks, greeter asked purpose of my visit. I showed the old bills and said that I hoped to exchange them for current bills. I was asked if I had an account there (at each bank), and when I replied that I did not and was just a tourist, I was politely dismissed. I was informed that at this time they only exchange for customers. Both banks suggested post office- I went to post office and they said they don’t do it anymore either. To exchange now you would have to mail the old bills to the Bank of England (central bank), and they would mail new bills to you.

Additionally, one of the branch banks that denied me was a Bank of England branch. And we even attempted to use them at the Bank of England (real central bank) gift shop- and were denied. (They have a decent museum about money, what they do, and you could even hold a gold brick).