You are very unlikely to input MEL to LHR on the QF website and find availability. BUT that doesn't mean it can't be done...

Do you have the flexibility to book close to the departure date?

You may find Emirates availability and JAL availability (ex SYD via HND) a week or two before departure. BUT Emirates redemptions have insane fuel charges thanks to QF's gouging of cash copayments, unless you know the tricks to route to avoid them....

Picking up on some concepts above.

Search in sectors - Australia to Asia and then Asia to Europe. Then compile your itinerary using the multi city function on the QF website, which is absolutely hopeless at stringing otters two for more flight sectors o note one search).

If necessary chunk the itinerary across bookings. A paid daytime flight on Scoot or JQ may be survivable to an Asian city, if you are interested in a stopover.

Remember that QF partners with Air France (search from Asia to Paris) and KLM (search from Asia to Amsterdam) and Cathay (via Hong Kong) and JAL (via Tokyo).

In the meantime keep saving those points (hint Qantas wine).

And diversify into Virgin Velocity to escalate your options.

Currently the most available an cost effective option is from SYD via HND to CDG on JAL with the overnight in HND (there are airport hotels), but that's one or two weeks out..

...patience...flexibility...