I'd heard that washroom wasn't universally understood in the US.

Nope. My church youth group volunteered at a summer camp in Ontario several years ago. On our first day there, a Canadian girl with a grimy young camper in tow asked me if there was a washroom in the cafeteria building. I had no idea that she meant a bathroom, and said "Well, there's a sink in the kitchen..." I got a very strange look, as you might imagine.