Just not funny. I admit, I am British and I like the Royals, but I'm not above laughing at them. Let's accept it, they are prime targets for satire and humour. Spitting Image and Family Guy have had some pretty funny jokes aimed at the Royals in the past, the subject matter is not the problem, the execution is.
It is like watching an extremely unfunny Family Guy Special. All you have to do is get rid of the cutaway gags, the sarcasm, and get a low budget Stewie Griffin. Then triple the poop jokes and add poor British accents and you have The Prince. I'm sure that it will appeal to some 15-year-olds, it's just difficult to see who HBO's target audience was... Teens enjoy poop Jokes and swearing but don't care about the Royal Family, adults may be interested in Royals but tend not to find poop hysterical.
The other big "jokes" are the Queen swears (wow, that is cutting edge stuff), Prince Harry is a bit dim and separated from reality (no one strained anything coming up with that), and Prince Charles is a mummy's boy who wants to be King. It just is all very obvious. The only likeable character is Meghan, unsurprising given that she is the only American. However, even her presence feels like a missed opportunity, given that she is widely perceived to have climbed the ladder of society on her back, yet no mention is made of this. They have made the characters too 2-D and predictable, predictable comedy can't be funny. You know roughly what the characters are going to say before they say it.
My other complaint is that it mocks people who can't defend themselves. To base the series on the life of a real child was a weird move, and to include Prince Philip after he's dead just comes across as misjudged. Even the moments that are vaguely funny involving these characters you can't enjoy because you remember that you are laughing at a real life child and a good man who is recently dead.
I'm sure someone, somewhere will like it, I just hope I never have to meet them. All in all, I have had better nights in, including the night my appendix burst.