The monarch can suspend parliament. The monarch can dismiss a prime minister (not happened for > 100 years). The monarch can appoint a new one. Every law has to be signed by the monarch (it's not been refused in 300 years). The monarch is commander in chief.
Power resides where people believe it resides. The monarch can only do any of that if people believe in the monarch.
The monarch can do all those things, but in reality if she tried to dismiss a PM or refuse to sign a bill, it would be a matter of days or weeks before the monarchy ceased to be an institution.
The monarch can dismiss a prime minister (not happened for > 100 years).
Actually, in Australia the Queen's representative (the Governor General) did dismiss the Prime Minister[1]. Interestingly, the Governor General position is closer to a ceremonial president in many other places in that they are appointed by the PM. If the PM at the time had acted to sack the Governor General that would have been just as legal as the other way around.
(Actually there are some good arguments that sacking the GG would have been "more" legal, in that the GG is supposed to act on the PM's advice. Fortunately the PM didn't push the issue, because that is the road to real trouble)
Power resides where people believe it resides. The monarch can only do any of that if people believe in the monarch.