From my understanding, the colonies each had their own parliament with power over internal affairs, but they could by overrulled by the imperial parliament, their govenors, or the colonial secretary. This would have happened quite rarely, though, as Australia was wealthy and loyal, and no British government would have wanted to disrupt the flow of gold and wool from their distant colony.
Each colony had a different constitution, traditions and political culture, so if your going to do a TL on colonial Australia, your going to have to research quite a bit. Try looking up the hansard for the colonial parliaments to get an idea of what kind of things they were dealing with and the way they dealt with them.