I've studied the subject a little bit ( I was a big WCW fan in my early days, and my grandpa was an avid fan from the territory days ). I can't speak too much about the UK, but I can answer your question about the US.

Amateur wrestling ( as we know it ) has it roots in old folk customs. Our wrestling came from European traditions, which eventually evolved into Greco-Roman, collegiate, and freestyle wrestling.

What made it big in the US was carnivals. Wrestling was often a sideshow attraction, which after a while the wrestlers would travel around and defend their titles. They started "fixing" the matches in the 1920s, which lead to a larger split between amateur/collegiate wrestling ( sport ) and professional wrestling ( entertainment ).

It's not much info, but it's all I got