The thing about Britain and England before it, is the structure of the relationship between Parliament and the Monarch isn’t subject to the legalistic “gotcha” games you guys are speculating on. In reality when the conflicts between the two have reached “certain points” of irreconcilability in the past, they’ve been settled by force. Typically in Britain and England before her, the monarch lost. This is all the way back to some rebellions by the barons that resulted in a King signing Magna Carta.
By saying it’s “settled by force” in the past this meant actual armed force. Most likely by the time of the 19th century and later, it wouldn’t be so dramatic. Parliament would simply assume powers it hadn’t assumed before, pass laws removing a distasteful monarch (as it had basically done before) and that’d be the end of it. It isn’t too different from how James II’s reign ended, excepting he physically ran from London (throwing the Great Seal of the Realm into the Thames) probably in fear of ending up on trial and short a head ala Charles I. That gave Parliament the legal path of saying he had forfeited the throne. It’d have to use a slightly different legal rationale to remove Edward VIII.
If it happened today, I think Britain would probably establish some alternative to the monarchy. I think they’d create a new figure head, non-hereditary Head of State position. If it happened back with Edward VIII my opinion based on politics and public sentiment of the time they’d have simply removed Edward and put his brother George on the throne.
Now, if a PM tried to tell the Queen that she couldn’t wear hats, then this isn’t how it would go down. Instead there would be widespread political outrage over the PM behaving basically like an imbecile and a crazy person, and he’d likely be voted out of office by the Commons and new elections would be held. But aside from some bizarre ministerial overreach the public and the commons would side with the PM. The Westminster system has some nice features when it comes to dealing with stupidity. Namely a PM that goes off reservation can be made not-PM pretty damn quickly. He wields a ton of power while he’s PM, but unlike say the American President who is nigh-impossible to remove from office for misbehavior a PM can be done in a few hours.
Historically there were a few situations where Parliament and the King couldn’t reconcile their differences, but it never reached a situation where the King was removed. During the reign of Charles II he eventually lost support in Parliament and could no longer form a government to his liking, he dissolved Parliament (they continued operating while conducting impeachment proceedings against a crown favorite, but otherwise assented to the dissolution) and the next Parliament to be seated was even more hostile to him. There were long stretches where the King couldn’t finance wars or other things he was interested in due to lack of support in Parliament, and in a sense there was a 17th century version of gridlock. The King actually had some revenue independent of Parliament, in the form of long standing rights to collect certain levies or tolls on certain economic activities, and his personal holdings generated income as well. Charles was able to keep some of his Kingdom running off of that, but it wasn’t enough to cover everything he wanted.
He was the last King with sufficient income independent of Parliament to operate at all, by the time of George III (who had multiple conflicts with Parliament himself) the monarchy was totally dependent on Parliament even for the maintenance of their personal household. Funding even a small part of the government would’ve been impossible by that point in time. George III actually essentially blocked Prime Ministers that he would’ve found distasteful during his reign, and he had stretches where his chosen PM was largely powerless in Parliament.