Does the reigning monarch actually have the power to appoint the PM, or does the reigning monarch have to wait until the majority coalition elects a leader?
Elizabeth II is queen of Canada also - I recall when the Liberals elected John Turner as leader, Trudeau the Elder stepped down. I doubt the queen could have overriden the party’s wishes.
The Queen has a lot of interesting “theoretical” powers. But as a matter of practical reality no monarch since William IV has appointed a PM without the support of the Commons.
In Canada the Queen is ceremonially represented by the Governor General. His duties are defined in the Canadian Constitution, which unlike the British, is actually written down.