The story is set in a very near future, in which Queen Elizabeth II has left this Earthly existence, and her son Charles, long the heir apparent Prince of Wales, has ascended to the throne at last. This Charles is much like the Charles we know - a liberal progressive, who values freedom of the press to the extent that he refuses to sign a law passed by Parliament which would restrict it. This is something that Elizabeth didn't do - refusing to ascent to the will of Parliament - where the fun aka exquisite drama begins.
I won't tell you exactly what happens, other than that King Charles' actions unleash a crisis of government indeed in the UK, and that Charles, William, Kate, Harry, Camilla, all royals behave in ways that seem consistent with what we know of them in our reality.
Charles, in particular, comes across as a monarch who wants his reign to be meaningful - that is, make a difference, do the right thing as he sees it - but is ultimately dependent upon his family. He is, in many respects, a perfect Shakespearean hero, caught in the cross-hairs of conflicts impossible to resolve.
Speaking of Shakespeare, Diana appears as a ghost who speaks (separately) to both Charles and William, and the dialogue is delivered in blank verse, with occasional and highly effective rhymes. The acting is just superb, with the recently deceased Tim Pigott-Smith in the performance of his career as Charles, and excellent work by everyone in the royal family and beyond.
I can't imagine what the real Charles and William and Kate thought of this. I hope they recognized it as the transcendent work of genius that it is, if even only to their inner selves. We the public have no such conflicts. Writer Michael Bartlett and Director Rupert Goold deserve every applicable award, as do the leading members of the cast, especially Pigott-Smith (as Charles) and Oliver Chris (as William) and Charlotte Riley (as Kate).
In our real world, in which Trump in the White House seems like an alternate reality too absurd to believe, King Charles III as near-future alternate reality is somehow as satisfying as it is deeply disturbing.