It has actually happened during the Queen's reign in 1974, she commissioned Harold Wilson Prime Minister even though he didn't have a majority. In 2010, David Cameron, and it's the situation now as well where she recommissioned Theresa May without a majority.
In all cases, they had the basic support of enough MPs outside their party to hold on. But at any time, the House of Commons could have chosen to have a vote of no confidence and remove them and in that case, the Queen would have to appoint a new Prime Minister who had the confidence of the House, or call an election - in practice, the last act of the outgoing Prime Minister should be to ask the Queen for an election, rather than put her in the position of having to act without his or her advice.
The type of scenario I think you are talking about here is one that would be very unlikely to arise except in a situation of crisis or on a temporary basis until an election can be held (we had this situation in Australia in 1975, which shares the same basic system - the Governor General faced with a Prime Minister (Gough Whitlam) who was trying to act illegally in the face of the Senate (our upper house) blocking his budget, sacked the Prime Minister and the entire government and commissioned the leader of the opposition, Malcolm Fraser, as Prime Minister, even though he did not have a majority in the House. But he did so on the condition that Fraser would act as caretaker Prime Minister only until an election could be held (which the Fraser lead coalition went on to win in a landslide confirming him in the Prime Ministership, which he held for seven years after that). But that was to resolve a constitutional crisis that a competent Prime Minister would not have allowed to develop. It really shouldn't happen. In Fraser's case, the House of Representatives did immediately pass a motion of no confidence in him, but as he'd already advised the Governor General to dissolve Parliament and call an election at that point, it was largely moot.
In a true crisis situation in the UK, the House of Commons might well accept a Prime Minister without a majority appointed by the Queen at least until the crisis was dealt with - as an extraordinary situation requiring an extraordinary solution - the precise reason the Reserve Powers of the Crown still exist. For use, basically in an emergency.
And what would happen if she didn’t commit the armed forces to something already approved by Parliament?
The Army is actually under the control of Parliament (although the Queen is Commander-in-chief) - that's why it's the British Army, not the Royal Army (as the Royal Navy and Royal Air Force). But again, unless there was some extraordinary reason, the Queen would not do this, and if there was an extraordinary reason, in the end, it would come down to who the armed forces chose to obey, and why, and who the people chose to support. If the Queen was blocking actions of a tyrannical government out of control, it could work. If she wasn't, it would likely lead to the end of the monarchy and a British republic.
The constitutional monarchy is based on the idea that a constitutional monarch acts to protect the constitution of the nation. A good government shouldn't make that necessary.
Thanks for the lesson in British government.
Thanks for the info. We Yanks are quite confused as to what the Queen can and cannot do. From your post, it seems like the Queen is the last line of defense against an illegal act (and - no offense meant - that’s about the extent of her powers).