This is going to be scrapped, if hasn't already but the fastest route to Ads is
Qantas : Heathrow - Bangkok - Syd - Ads. Off the plane in Bangkok for approx 1 1/2 hours, land in Syd about 6am pick up bags, clear customs immigration etc go to Qantas transfer give them your bags and they bus you per to the Domestic terminal and your flight to Ads leaves about 8.30. Actual time in air is about 20 hrs. British Airways will do this route to Syd so may still be possible.
The other thing is to fly by whichever airline to Singapore and stop over for a day or two, it works wonders in acclimatising to the bright light and heat which may meet you if you arrive in Aus in summer. Singapore is also only two to two half hours time diff so you can take some time clearing the jet lag.