If Brown had refused to leave office, then it would be up to Parliament to resolve the crisis. They would have voted to strip him of his post of Prime Minister, and then ordered law enforcement to take him into custody if he still refused to go quietly. The Queen would almost certainly not have done anything (except perhaps for privately advising the PM to observe the will of the people) while this all played out.
This is certainly a possibility - to allow the Prime Minister to continue in office until the House of Commons convenes and then allow the Commons the opportunity to pass a motion of no confidence in the Prime Minister.
But the problem with that solution is that in response to such a motion, a recalcitrant Prime Minister could choose to advise another general election, rather than resign.
Britain cannot afford such a situation to develop, and I do not believe Her Majesty would allow it to develop.
Instead I believe she would take a somewhat similar approach to that taken by her Australian Governor General Sir John Kerr in 1975, where he spoke to the Leader of the Opposition and after obtaining reassurances from him needed to resolve the crisis, dismissed Prime Minister Gough Whitlam and commissioned Malcolm Fraser as Prime Minister. There would be one important difference - having just fought an election that gave them a majority in the House, Prime Minister Cameron and Deputy Prime Minister Clegg would not be obliged to ask for an immediate general election, a necessity in Australia in 1975.
The only impediment to this would be if the Liberal Democrats refused to serve in such a coalition, and I could certainly see that happening. In that case, I still believe Her Majesty would be likely to commission David Cameron as caretaker Prime Minister on condition he immediately advise an election - this would create fundamentally the same circumstance as allowing a recalcitrant Gordon Brown to advise on in the wake of a no-confidence motion, but without the time and expense waste of an aborted Parliamentary session.
While the Queen may theoretically have the power to remove a Prime Minister, in reality, the Prime Minister serves at the will of Parliament. Only if the entire democratic process breaks down -- as was happening in Spain under Juan Carlos when he made his public appeal -- would the Queen dare to intervene in a crisis. All other legal and parliamentary avenues would have had to have been exhausted first.
I believe the best current precedent on this is the Australian constitutional crisis of 1975, and that therefore sets the current 'limit' on how severe a crisis would need to be before the Head of State intervenes. And that crisis was not as serious as a Prime Minister who had lost an election refusing to resign - Whitlam merely couldn't pass a budget.
The bottom line is that the monarchy is in a "use them *and* lose them" situation when it comes to its constitutional powers. To all intents and purposes it is a ceremonial monarchy with no real power because if they try to interfere in any way with the democratic process, they will be slapped down, possibly to the point of annihilation. (And that's as it should be.)
This type of circumstance would only develop if a Prime Minister was acting completely unreasonably and dangerously.
In such a circumstance I believe Her Majesty would risk both her own role and the Monarchy itself. It is her duty to protect her nation and she would risk all to do that.
Fortunately it does not arise as Gordon Brown has acted within convention, and has not even come close to allowing such a crisis to develop - as I would hope any Prime Minister would. He pushed things a bit, perhaps, but he didn't push them to anywhere near their limit, and certainly not to breaking point.
It would be an extreme situation where the Queen had to intervene - but if it reached that stage, she would, in my view, do so decisively.
I didn’t know about the Australian incident-—interesting-—but having read up on it I see that the Queen was not actually involved in it at all, and that Kerr, the Governor-General, while officially appointed by the Queen, and nominally her representative in Australia, was in reality selected for the post by the Australian PM (with the agreement of the leader of the opposition) and made all the decisions himself.
It’s interesting that Kerr was afraid that the PM would go over his head and ask the Queen to dismiss him before he had the chance to dismiss the PM-—that *would* have embroiled the Queen in a very nasty constitutional mess-—but in the end it was resolved only by those who were either elected by Australians or appointed by leaders who were elected by Australians, and I think that’s qualitatively different from having an unelected, unappointed monarch involved making constitutional decisions. And things have changed a lot in 35 years.
As it was, there were strong calls for Australia to become a constitutional republic in the aftermath, though they eventually didn’t amount to anything.
I really think that unless there was a total impasse, the Queen would have to stay out of it, with the only exception maybe being a private phone call to encourage the PM to do the right thing. Even in extremis, I suspect the only action of the Queen would be to accede to the actions recommended by Parliament to have him forcibly removed from office.
Thus I still believe that any direct intervention from the monarch would have to be from a far greater crisis than someone squatting in Number 10. A successful Gunpowder Plot (Heaven forbid) or a military coup would certainly qualify.
As for Gordon Brown pushing it, I disagree. He was right to stay on as PM while the negotiations were going on, (you can’t have a power vacuum and Cameron was not in a position to claim a workable majority) and he was perfectly within his rights to pursue a coalition with Clegg before Cameron and Clegg had reached a deal. If Labour had united behind him in reaching a deal with the LimDems, they would have had more MPs than the Tories, and would probably have had the backing of the nationalist parties too-—enough to win a vote on the Queen’s Speech. It would have sucked for the Conservatives, sure, but there would have been nothing they could have done about it.
In any event, it didn’t happen and a day later, once it became clear that the Tories had reached a deal with the LibDems, and Brown resigned almost immediately.
All in all, I think we’ve seen British democracy at its finest in these last few days. Putting all the party politicking and backroom deal making to one side, we just saw a peaceful, orderly transition of power five days after an election that threw up an result with no clear winner, and very few precedents. There was no civil unrest, no talk of military intervention, no attempts to thwart the constitution, and the like. And while that’s exactly how one would expect British grown-ups to behave, if you think about it, there aren’t that many countries or times in history where such a thing was likely, or even possible. So I would give credit to everyone involved, even Gordon Brown on this one.