"Oz-tray-lee-ah"

Edited to reflect a pronunciation, not necessarily a correct or accurate one, from which it's easy to derive 'oz'.

It's almost impossible to produce "oz-tr" because the t is unvoiced. But as I mention sidethread, the term for a person, Aussie, is pronounced with a /z/, and the Oz pronunciation is probably related to that.

They actually pronounce it “straya”.

...and nobody says "oz" when they mean to say "ounce". These are literal liberties, which fit either way, I think.

Voicing s as /z/ is widely variational. For example, I voice the s in "us", and "Muslim", while others don't. While I don't voice the s in Australia (/ɒ.ˈstreɪ̯.li.jə/ ~ /ə.ˈstreɪ̯.li.jə/), I do in Oslo (pronouncing it /ˈɒz.loʊ̯/).

They do say it with a /z/ sound. That's why it gets spelled Oz.

Compare Aussie, which is also pronounced with a /z/.

I've never heard anyone pronounce 'Australia' as 'Auztralia'. 'Aussie' as 'ozzie' yes, but never Australia.

I'm not sure where exactly I heard this, but some time when I was young, I either was told or read somewhere that the author came up with the name "Oz" when looking at a filing cabinet; the top was labeled "A-N", and the bottom was labeled "O-Z"

Because oz sounds like "aus"