Don't Australia have its own head of state besides the queen?
Here in Canada, the topmost position is supposedly the queen, but the Prime Minister is the actual head of state. The monarch has no control over the country except for picking a Governor General and that's on the "recommendation" of the Prime Minister (if I remember correctly). The Governor General does little except give out awards and spend taxpayers money on travel etc anyways.
I think that not having the monarchy would mean that Australia would have moved on from the history of the Monarchy and the colony, but I agree with Einherjar that it's not a pressing issue, as the monarchy of today are different than back then anyhow and there are more important things to consider.
Still having a queen in Britain might help to not strain relations between the two country...