I guess not totally pointless, Australia would get radiated somewhat from drifting fallout, and I guess there is something interesting in the idea that the whole world didn't fall at once, history proceeded for a while before coming to a more climactic end shortly thereafter.
But it just feels conceptually like a bit of a hat on a hat, and also there's the problem that in Beyond Thunder Dome Australia didn't just receive drifting fallout, it was nuked directly: Sydney was destroyed by nukes, and I believe that it was nukes that brought down the plane that the child tribe came from. If things in the interior of Australia had already devolved into the neotribal Gigarape-and-Carnage Rock n' Roll BDSM Motorocalypse, how could there have been anything worth nuking left in Sydney? There are some ways that can be made to fit, but it's rather ungainly and it's cleanest to just say the total social collapse we see in Road Warrior came some time after both Peak Oil and a nuclear exchange.
You're right, evidence of nuclear war is mostly confined to Beyond Thunder Dome, though perhaps with one exception - I do recall Lord Humungous having weird growths or tumors or something despite otherwise being in excellent physical shape, which I always took to be from radiation. But otherwise there's not much evidence, and perhaps I'm misremembering, so perhaps that was Miller's original intention.