I’m in the reverse situation in that I moved from Oz to the UK, albeit for a longer (indefinite) period. If you are already a BAEC member then I would not consider joining QFF (even for domestic flights). It is a poor frequent flyer programme. (That said, I am sticking with QFF for another 12 months to make Life Time Gold).
Adding your BAEC number to Qantas bookings will allow you to earn Avios and Tier Points, and you can earn and redeem Avios for flights in Australia (or another Oneworld partner). You will also be able to use the Qantas mobile app with your BAEC number added without being a Qantas Frequent Flyer.
The good thing about BAEC (and there are getting fewer) is you are not penalised with Tier points when you fly partner airlines, only reduced Avios. So you need to decide what is more important for you, getting Avios or moving from BAEC Blue > Bronze > Silver > Gold.
If you want to sign up for a different frequent flyer programme, then Dave Noble has provided a better assessment of American Airlines than I could.