For a start, there are all the classics. Yes, it can be a little bit weird to hear someone singing about winter and snow when you’re in the middle of a heatwave and want nothing more than to go to the beach, but we do have such songs. The cognitive dissonance is something you get used to.
You then have a number of songs which are Christmas carols but specifically have an Australian “vibe” to them. “The Carol of the Birds” (often known by its first line “Out on the plains, the brolgas are dancing”) is a very attractive example of the form, but there are a few others – “Across the plains one Christmas night” also comes to mind, although that’s more due to the traditional joke about the name of the third drover’s horse.
There are also various attempts at re-doing the classics for an Australian audience. There was, for example, an ad a couple of years ago (which, now I think of it, may not have been for an Australian company) featuring Ronan Keating singing about a “Summer wonderland”. Doing some Christmas shopping the other day, I heard some performers singing something which ran “We wish you a ripper Christmas and a cracker New Year”, or so, which was…interesting.