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.

Thanks for your comment. I don't know how ordinary US Americans refer to Australia and whether they use 'Oz' to do so but I am not surprised to hear from you that they generally don't. What I was talking about, and where I should have been more specific, is how Australia is referred to in the US media where I have quite often seen it called 'Oz' (and perhaps for the mundane reason of making for a shorter headline). This isn't nearly so common in the Australian media.

Btw, I agree with you that US Americans often mispronounce 'Aussie', but I can understand why, if they encountered the word in print before hearing it spoken.