At this time (as in this generation) there is no reason for Australia, NZ and Canada to be anything but Republics.
Not long ago there was 'no border' (so to speak) where there was a sort of free trade and immigration/emigration and in real meaningful way were were all citizens of the same Kingdom.
An Australian could have simply moved and worked in London and been given same rights, services, protections etc.
Same back and forth all over,
but,
All that is gone now. A Canadian could no more go move and work in the UK than anyone else outside the EU,
but,
A German could.
An Australian will have to go through UK customs and be no more given any special rights, benefits or protections than would Chinese or Russians.
Oh there may still be a handful of 'commonwealth' sharing and ties. 'grandfather clauses' or some legacy ventures here and there,
but,
realistically, Australia would just benefit their country and culture. Canada would too. If anything ties and bonds between them and the UK would be as strong or even stronger.
It comes down to minting the coins.