Don’t get your knickers in a twist, mate! I know that technically, there is no such thing as a “British” accent. But what else would you call the English-ish dialect that’s long been used as a catchall in American-made movies and TV shows set anywhere ancient, foreign, and/or magical?
The fantasy thing, at least, makes a certain amount of sense. That genre has its roots in stories like J. R. R. Tolkien’s Lord of the Rings saga and C.S. Lewis’s Chronicles of Narnia; naturally, those authors’ homelands (and, therefore, their way of speaking) are associated with the stories their work has inspired.
Perhaps more importantly, made-up fantasy lands tend to have cultures and customs based at least partially on those of medieval Europe – and even if people in medieval Europe didn’t sound remotely like denizens of present-day London, using modern British inflections makes fantasy characters’ speech sound more authentic to most people’s untrained ears. Hearing knights and princesses speak like they were brought up in Cincinnati just seems wrong — possibly also because the United States has existed for only 237 years.