Um...'fanny' here in England is a slang term for what is also called a girl's 'front bottom' or in less coy terms, vagina. How it got moved to the back bottom over there I don't know LOL Here it was a 19th Century coined term but the exact origin/association is unknown. I would imagine it was here in the 1800's and probably got shared amongst the international Allied troops of WWI so that by the mid 1920's it would be introduced on the American scene. It would probably be well-established by the 1930's and on the wane, dying out by the 1940's.
I imagine the move from vagina to buttock was one of simple misunderstanding. Sort of like Squaw - it actually means 'penis' but white misunderstanding means that people routinely make the mistake of calling me a 'squaw' when I'm just a mixed-blood Native American woman and like women everywhere, completely lack that appendage ;o)