Do British people make fun of Americans in American accents?

Yes, as there are people everywhere that immitate countrys accents and make fun of them.

The British have a long history of poking fun at almost everything. For an allegedly pompous race they have a remarkable sense of humour, much funnier than American slapstick. The British, unlike Americans, are happy to laugh at themselves, and the British will poke fun at Americans as happily as they poke fun at themselves. Of course, Americans give British comedians a lot to work with.