OP, do you want to keep earning BA Avios points? If you had to pay some $/Gbp along the route. If you have enough points for all the way, without paying any more, lucky! In which case, its your choice if you go BA/QF. Or EK, if you can get 2 reward seats with them.

Or UK to Dubai with BA, if they still fly there, and then EK with a QF flight number to Aus.

I would think it would be a pain in the bum to start earning points with EK's FF system from scratch, my point of view, so maybe, just go with BA to Sin, and then BA/QF from Sin to Aus. Using points from the UK to Sin in J, and then from Sin and if enough points, BA again to Syd. Or paying $/Gbp from Sin to Aus flying QF.