Richard and I saw this play on Broadway on 11/17/15. The author, Mike Bartlett, calls it "a future history play" - - the play opens with the funeral of QE II. Charles is King, William is the Prince of Wales, Kate is the Princess of Wales. Strangely enough, no discussion of Camilla and her role. She was sort of the comic relief, but no one talked about her title.
The most remarkable thing about the play is that it's written in verse. Not rhymed verse, but metered verse. Beautifully written, and the actors (only one American actor in the cast) had a wonderful facility with it. The greatest performance was by Tim Pigott-Smith as Charles. He was supreme, he effortlessly projected the outward authority and the inner doubts of the man who waited so long to be King. He was especially good in the late, Lear-ish section of the play.
My other favorite performance was by Lydia Wilson as Kate. She had the most thrilling moment of the play, a soliloquy halfway through the second act where she says (I'm paraphrasing) that everyone thinks she's pretty and gracious and they don't pay much attention to her, but she has an agenda and she will further that agenda. You don't realize it at the time, but it's the turning point of the drama and everything that happens after it was engineered by her. So the demise of King Lear is brought about by Lady Macbeth. What more do you want?