The ever-reliable TVTropes has a small section on the Royals in politics at the end of its British Political System
Mostly, however, the monarch just drinks tea and advises the Prime Minister. Several prime ministers, among them Margaret Thatcher, have attested that this is typically not just ceremony: the Queen has access to most significant government documents, and apparently, has spent several hours a day every day for the last fifty or so years going through them. There's very little she doesn't know about government policy.
While the monarch does technically have the ability to veto any act of Parliament, to refuse a nominated Prime Minister, to sack the Prime Minister if he messes up, or to mobilise the army, to actually do so would likely cause a massive public outcry as it would be going against the will of the people by virtue of going against their democratically-elected leader. It is also extremely unlikely that the monarch could face-off against the rest of the British political establishment and win.
Charles I tried tangling with Parliament, and ended up...well, several inches shorter, and not in need of any hats for the foreseeable future. The monarchy these days, by and large, know better than to even attempt it.
However, the current Queen does have one useful function. As the armed forces swear allegiance to the monarch and not to the Government, should the Prime Minister declare himself a dictatorial leader the queen can directly order them to stand down and, if necessary, turn against Westminster. This would be an awesome ending to a film, if anyone wants to make it.
Yes. Yes, it would.