Which of the following sentences is good English?
(1) I am a local born Australian (as opposed to a naturalised Australian)
(2) I am a locally born Australian (as opposed to a naturalized Australian)
(3) I am a native born Australian (as opposed to a naturalised Australian)
(4) I am a native Australian (as opposed to a naturalized Australian)
My suggestion is #3. Australia is a big country. You may have been born in Australia, but thousands of kilometers from where you're standing, so it wasn't local.
Actually, my suggestion is really "I was born in Australia." It's completely clear.
My problem with "native Australian" is that "Native American" means that you're descended from the people who lived in the Americas before European colonization. I don't know if the same phrasing applies in Australis, but my first thought is that it would be you are an aborigine.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.