Having vague ideas of Parliamentary supremacy, I’m struck by reports that Boris Johnson could simply ignore a vote of no-confidence, and remain office to push through a No Deal Brexit, even in the face of legislation prohibiting such a course of action. As far as I can tell, these arguments are based on the premise that Parliament must nominate an alternative, and the assumption that neither Corbyn nor anyone else would prove acceptable to a majority. (Update: On more careful reading of the linked article I see that the new boss, Cummings, saying that Johnson could, for which I read would, ignore a vote for a new PM).
That might be true, and then again it might not. The question that occurs to me is whether Johnson could also ignore a vote in favour of a new PM, and if so, what could be done about it? One possibility is that the Queen could dismiss him, and invite the new PM to form a government, which would presumably hold immediate elections.
That, pretty much, is what happened in Australia in 1975, though the government’s position was far more tenable than in the hypothetical that I’ve outlined above. The government had a majority in the House of Representatives (our equivalent of the Commons). However, the government was unpopular and the Senate (similar to the US Senate in most respects) refused Supply, creating a financial crisis. The Governor-General (representing the Queen) sacked the PM (Whitlam) and installed the Leader of the Opposition (Fraser) in his place. Fraser called an election and won.
I honestly have no idea whether if Johnston refused to go, he could be removed, by the monarch or otherwise. For that matter, could he get the Queen to prorogue Parliament indefinitely, and govern by decree? That sounds inconceivable, but maybe only in the sense of Vizzini in the Princess Bride.
It’s worth noting that the absence of a written constitution isn’t the critical issue here, or at least not to the extent often claimed. Australia has a constitution, but it’s silent on all the relevant issues (it doesn’t even mention the office of a PM).
That’s enough from me. Useful links and informed comments much appreciated.