Why would changing the rules of succession affect Australia?

Because for each of the 16 countries that King or Queen is also the King or Queen, not by default but by that Nations laws. If the law changes for England that effects only England and we need to change our own laws for it to be binding on us. Otherwise you will end up with Queen Martha of England (as the first born child) but we will have King Trevor of Australia (or Canada or any other commonwealth country) because our laws still call for the first born son.