Airlines seem to have a habit of hanging on to UK Air Passenger Duty when a passenger doesn't travel, probably because they've already handed it over to HMRC and they might have trouble getting it back themselves. Even if you'd cancelled the flight yourself (or simply not bothered to turn up at the airport), you'd still have the right to claim APD back. [Where it's
your fault that you didn't travel some airlines charge an admin fee which means that it's not worth claiming APD anyway though but they shouldn't be applying an admin fee when they've cancelled the flight].
So, assuming that the 'taxes' referred to are in fact APD (and not something else, such as charges levied at your intended point of arrival in Oz), it seems to me that you should be able to get those taxes back.
All I can sugget though is hassle, hassle, hassle! Try using this web form:
https://qantas.custhelp.com/app/ask/c/606