In a recent interview with Shortlist (via What Culture!) Refn reveals, "I would have loved to direct 'Doctor Who' but they didn’t want me — they turned me down last year. Maybe if they revive [1970s sci-fi series] 'Blakes 7' I could do that. I love it. It’s great. That could be fun to update."
Of course, this likely before he won Best Director at Cannes and "Drive" became a buzzworthy movie. For all the producers knew, he was just some random Danish director who didn't know how to drive who was calling them up asking to direct their critically acclaimed hit show. And it's unlikely he would've made his stamp known on the show in any major way (showrunners don't like to deviate too much from the established tone and form), but it still could've been pretty interesting.
At any rate, he might've been more inspired than he was when he took on the 'Nemesis' gig which he admits was purely for the cash. "I did two episodes of 'Marple' back to back because, basically, I needed the money. It was fun — a good experience. I really liked working with Geraldine McEwan, she was a great actress. And Amanda Burton, too," he said.
Anyway, we'll have to leave a Refn directed "Doctor Who" to the imagination, but don't fear. He's currently shooting "Only God Forgives" with his boytoy Ryan Gosling, so hopefully we'll see it some form before the year is out.