As I said, I'd never heard it before you used it in this post. I've never seen it written in print or said on radio or television. Perhaps it's still industry jargon that hasn't been used by the media or used offhandedly by an actor. But I don't believe it for the above reason and because it is just such a British-y term.
Any evidence that its origin is American?
At the link:
"Showrunner" is a term originating in the United States television industry referring to the person who is responsible for the day-to-day operation of a television series"
It discusses the origin of the term, and clearly states that it originated in the United States and came to the UK only in the mid-2000s with DOCTOR WHO and the MY FAMILY comedy series (the latter of which used an American-style writing room and even American writers, unlike most UK sitcoms which are written by just one person).