How about a Bitstamp? Bitstamp is a British cryptocurrency exchange, isn't it? So this should support Faster Payments scheme in the UK. I am not a citizen of UK, therefore, I can not share my personal experience. But you can read the reviews about this cryptocurrency exchange
https://revain.org/exchanges/bitstamp. I hope you find the answer to your question.
Bitstamp are Slovenian and always have been. They used to rent a mailbox in the UK but that's as far as their Britishness goes.
I don't think they've integrated GBP and faster payments yet but they have announced it. I'd still never use them for their disgusting invasiveness and evasiveness.