I notice that the dialogue of Narnian characters in the Narnia books sounds both distinctly dated and distinctly British. But I think there's no way to avoid that. Books written nowadays in America will have characters in fantasy worlds that sound like they're from 2010's America, and it will be very noticeable 70 years from now.
Like, where do you draw the line? Do you full-on write in Middle English?
Whanne that April with his shoures sote
The droughte of March hath perced to the rote.
Not sure the modern reader will stick around for a whole novel written that way, but maybe