Loads of threads on getting Oz availability if you do a search.
Fundamentally, odds are improved by a) booking as far as you can and b) being willing to spend a couple of days if necessary in BKK or HKG waiting for an available onward flight.
Even with a) you really need to ring up and book, say, out Monday and back Tuesday (on the day the Tuesday flight opens up) and then ring back 14 days later - if its a 2 week trip - to move your return date. If you wait until your return date opens up then the outbound will have probably gone.
There is a plan c) which is simply to pay cash if you can use the Amex voucher elsewhere. It is often possible to get J to Oz for £2,000 per person from Europe - not BA, but someone, often someone better than BA. At least this route doesn't give you availability problems. You need to keep an eye on other FT forums for this.
A quick play with the ITA tool for March 2001, for example, shows Etihad - an airline whose business class puts BA to shame, I'm led to understand - offering Sydney ex-Istanbul for £2,065 all-in, and of course you'd earn a stack of miles back as well.