Actually, when talking to an Aussie, we might say Adelaide, but to a non-Australian it becomes the full thing - Adelaide, South Australia, Australia.
When talking to an Australian we might say "Brizzie" (which we write as Brissie), and they would know what we mean. To any foreigner we would likely say "Australia, actually, Brizbn, Queensland", or "Australia, actually, a little place called Tallem Bend, South Australia". When explaining the birthplace of a family member I say "Osbourne Park, a suburb of Perth, Western Australia, Australia" (unless I am speaking with a fellow Aussie).
I was taught early on in my life to not expect everyone else to understand the shorthand in my brain, and to fully explain so there are to be no doubts - and that goes for place names with which I may be very familiar, but they may not.