I am not a recruiter for Cathay nor Qantas.
All of my mates who've got into Cathay have not got degrees.
They just had several thousand hours in piston twins and turbines ranging from Twin Otters, Bandeirantes, Brasilias and Dash 8s.
What you must realise is that you will not go from flight school to Cathay (or Qantas for that matter).
So pick a good school that will teach you properly as opposed a sausage factory.
If you are in Sydney, go and find out what tech courses are being run. They offer (or did when I was doing them) very good value courses.
If you want a degree, it will not hurt you. (It may give you a debt you don't need but it may give you some back up skills just in case). I just can't see the benefit of an aviation degree outside aviation. Most of my friends who have done them have done them once they're in an airline type job and what something to do to keep their brains working.
As you probably have learnt, there's no guarantees. Don't place all your eggs in the Cathay/Qantas basket.