At the moment, things wouldn't be terribly different if Oz were a major power, since the current PM's policies are fairly similar to your POTUS's. Except for the fact that the big money here is in minerals, not oil. Oh, and he cares even less about off-white people, whether indigenous or not. And he's more about expediency rather than making his own policy and sticking to it. See, it could be worse. Sort of.

NZ as a world power would be interesting. The current PM is a political scientist who specialises in foreign affairs. She actually knows where most countries are! (if not all) She's even interesting in negotiating with them! And she ain't no member of no old-boys' club, that's for sure.