"Keen on" isn't British. I use it often. It's just less popular than saying "fond of."

Example:
She was keen on finishing that book. - This makes it sound like she's more serious about doing it.
She was fond of finishing that book. - This makes it sound like she wants to, but will do it whenever. There's less determination here.

"Fond of" is just another way to say that you like something. "Keen on" implies a more serious liking towards something.