George Bernard Shaw was to the English language plays what Irving Berlin was to English language songs, good and prolific. When you're as prolific as both these guys were chances there will be a play or a song that won't be quite up to snuff. The Apple Cart goes most definitely in the category of the lesser work of George Bernard Shaw.
It certainly doesn't get much play in the USA and I think that one has to at the very least live in the United Kingdom to understand all the governmental references and just how the royal family fits into the structure of British politics. To the average American a lot of this will go right over their heads.
The Apple Cart was written in the late 20s and in one sense I think Shaw was a prophet. The King as played by Nigel Davenport was most likely based on Edward VII and also on the future Edward VIII. Already the current Prince of Wales then was giving cause for concern by his not sticking to the role laid down for him as British tradition evolved. Davenport's King Magnus certainly has the libido that the Edward VII had as witness by lovely young mistress Helen Mirren in her salad days and a wife a whole lot like Queen Alexandra. Acting honors go to Prunella Scales who plays the queen who like being queen with all the royal prerogatives.
You'll see some fine acting in The Applecart, but you can't truly enjoy a play that you have to have studied comparative government.