"Sneaker" has as many designations in English, too. I grew up calling them that or "tennis shoes". "Gym-shoes" was a word I found in older American literature and "athletic shoes" was a term found in formal notices. When I went to Britain for the first time, a friend made reference to my "trainers" and I had no idea what he meant.

By contrast, I can't think of any designation for a ham and cheese sandwich except "ham-and-cheese". I suppose "croque monsieur", but that's a very specific preparation--at least to me. A would-be bistro near work has this on its menu, but I was sorely disappointed when I ordered it since they used crusty rustic bread.