Hah. Australia's economy is built entirely around two things. 1. Extracting fossil fuels and minerals. 2. "services".

During a global catastrophe Australia would run out of fuel within a few months, fertilizer not long after that, and soon afterwards lose the ability to grow food let alone distribute it, or keep basic services running. The lights would go out because it has no capability or even wishful thinking to manufacture advanced electronics, precision parts, much in the way of plastics and chemicals to keep airplanes, trucks, cars, computers, communication, phones, power plants, transformers, railways, or tractors running. Basic antibiotics, anaesthetics, insulin, any kind of medical or surgical tools or technologies? Don't even bother. Most people who survived the initial waves of panic and looting and destruction would soon starve and freeze to death in the dark surrounded by coal mines and fields of wheat rotting where it lies.

Also, Australia is a strategic point in the southern Pacific and Indian oceans with submarine bases and communications, air bases, signals and radar stations. It was believed to be one of the countries targeted first by the Soviets after USA if it came to a nuclear exchange. The calculus might be a bit different with China today but Australia is not some neutral party that would escape unnoticed in the event of a world war.

I don't know what would be the safest place, but I know Australia is not it. Somewhere where people can actually build and make things, and whose people might be inclined to cooperate and be capable of enduring some hardship, would fare a much better chance.