My mother is Australian and my father is American. We (my two sisters and I) had a distinct Australian tinge to our speech which was obvious in elementary school. This caused me to actually miss test questions on syllabification in class, since a more typical US accent would say e.g. i-de-a but our mother broke it into i-dea. Mom actually protested that one to the teacher.
I've lost most of that except for a residual tendency to say eye-ther (either) and tomahto (tomato), which my Strine wife smirks at.
I'm Australian born and bred, lived here my entire life, and I say 'tomahto' and 'eye-ther'. I'm also firmly on the right-hand-side of the trap/bath split.
It's a mistake to think that there is a single Australian accent.
There's definitely a few different Aussie accents.
I live in central Melbourne, and most people I meet have an accent very similar to a Kiwi accent, close enough that after only a year since I came from NZ, most people think I'm an Aussie. The accent has been easy to pick up, just say 'feesh and cheeps' and use a high rising terminal (like you're asking a question), and you're good to go.
However, go to the outer suburbs, or rural Australia, or up the coast to NSW or QLD, and you get a whole array of different accents. You've got the ocker accents, the wog accent, the aboriginals have their own accent.
I've lost most of that except for a residual tendency to say eye-ther (either) and tomahto (tomato), which my Strine wife smirks at.