How did I get approval to use the British Royal Family as characters?

I didn't. But I don't expect that to be a problem. Charles, Camilla, William and Harry are all in the story of Future King. However, I didn't present any of them in a bad light. On the contrary, they're real people, even heroic figures. I also put a disclaimer in the front of the book that indicates Future King is entirely a work of fiction and, "the author intends no harm of any kind to anyone living or dead."

That's basically the same approach I took in my first novel which featured Walt Disney. And I never heard a squeak from the Mouse.