There are many to choose from! A.S. Byatt, yes, I love her too. But also David Mitchell, who's astonishingly versatile (and young) and Tom Stoppard as the best living playwright. If I spent another ten minutes, I'd come up with ten more, easy.
Certainly the book I've been most impressed by, and by some distance, of any writer currently living in the British Isles is Riddley Walker by the American, but long-term British resident, Russell Hoban. I must check out some of his more stuff; Pilgermann was also excellent. I admit to being very far from au fait with much contemporary lit.
Though I spose I've chosen to replace of the British Isles with in the British Isles. Maeve Binchy would have to be my revised choice.
The first name that immediately comes to my mind is William Trevor, for both his novels and his short stories (few do both so well).
I know he is no longer living, but I'm still going with Graham Greene. How's that for following the rules?
Well, Greene surely was a great writer, Scott. Deserved a Nobel.
What about Binchy? Surely she is as deserving.
I confess, Andrew, that I have not read Binchy, though I hear nothing but wonderful things about her.