I am from the US originally, and I literally never heard anyone call Australia "Oz", and didn't even know what it meant, until I moved to Australia. Americans — at least all the ones I encounter — are almost always confused by the designation (they think you're making a reference to The Wizard of Oz).

I think that's because "Oz" comes from "Aussie", which Americans mispronounce (they say something like "ossy" rather than "ozzy").

It's uncommon for Australians to say "Oz" in my experience, but it's still definitely a thing (albeit mostly in ads/marketing rather than daily speech). And even the ones who never say it still know what it means.

I don't know what people say in the UK because I have never been there.

:-)