The nicknames Oz or Down Under for Australia are terms much disliked by Australian journalists. I advise you not to use either one. Personally, I hate those awful expressions. It's worse than writing Xmas for Christmas.
Let me tell you why these little words matter so much. In serious journalism the name of the country is Australia. The people are Australians. The thing they do or make is Australian. That is the whole vocabulary you need, and it never lets you down. "Oz" and "Down Under" are travel-brochure words, the sort of thing an overseas sub-editor reaches for when they cannot be bothered learning the real name. When you use them you sound like a tourist, not a reporter.
Here is the test I apply. Would you open a hard news story about the United States by calling it "the land of Uncle Sam"? Would you call Britain "Blighty" in a court report? Of course not. You would be laughed out of the newsroom. Australia deserves exactly the same respect in your copy.
There is one narrow exception, and only one. If you are quoting somebody directly and they actually said "Oz", you leave their words alone, because a quote is sacred. The same goes for a proper name you cannot change, a band, a film or a festival that has the word in its title. Outside of those cases, write Australia every time.
It is a small discipline, but small disciplines are what separate the professional from the amateur. Get the name of your own country right, and right every time.