“but the United Kingdom no longer has any power here. The Queen does - as Queen of Australia - but that is separate to her role as Queen of the United Kingdom.”
Doesn’t the Queen still have the power to dissolve Parliament? I remember hearing something about that during a tour of the Australian Parliament years ago, although that is extremely unlikely to ever happen from what I was told as the Queen is generally hands-off when it comes to government affairs.
Yes, the Queen of Australia has the power to dissolve the Australian Parliament, and the Queen of the United Kingdom has the power to dissolve the British Parliament - in her separate constitutional role in each country.
It is very unlikely such a power would be used except on the advice of the Prime Minister but it is theoretically possible and it could be done in an emergency, for example, if a Prime Minister started to act outside of the constitution.
Within Australia, it's more likely such a decision would be taken by the Governor-General rather than the Queen (unless the Queen happened to be in the country) and it has come close to happening once - on November 11th, 1975. On that occasion, rather than dissolving the Parliament on his own behalf, the then Governor-General, Sir John Kerr, instead sacked the Prime Minister Gough Whitlam and commissioned the Leader of the Opposition Malcolm Fraser as caretaker Prime Minister on the express understanding that as soon as possible, Fraser would ask him to dissolve Parliament (which he did later that day), but it really was the Governor-General's call (and constitutionally the right one - the Prime Minister had been unable to get a budget passed and was on the verge of the government running out of money and his proposed solution was to illegally order the state owned Commonwealth Bank to loan the government money to continue - it wasn't quite unconstitutional but it would have been totally unlawful).
There is a bit of a strange gray area because the constitution delegates many powers directly to the Governor-General, including the power to dissolve parliament, while delegating others to or through the Queen. So there is some dispute about whether the Queen can actually exercise powers specifically assigned to the GG.
In any case it would be very unusual and there would probably be bigger things to worry about if the situation had deteriorated to the point of the Queen personally intervening.