I believe that BA will still have a daily through 744 (compared with a 772 now) stopping at SIN. As both their through flights to SYD, the only Australian destination served, leave within a few minutes of each other it will not make much difference to their offering. Presumably the ideas is that BA will still pick up traffic that is looking for a through flight. If a pax wants to stopover at BKK (or HKG) it does not matter if two airlines are involved.
QF will operate two daily through flights also via SIN with A380s. I am not certain of the origin in Oz, I would guess SYD & MEL.
The two airlines will be reducing their capacity on the Kangaroo route. For BA this is not great (a loss of a daily BKK - SYD flight offset by 3 x weekly additional LHR - HKG frequencies. 3 additional flights a week to GIG will soak up the freed resources. For QF it is a much more significant reduction of up to 14 weekly frequencies into LHR and they are talking of retiring four 747s.
Presumably the sixth freedom carriers such as EK wil pick up the traffic lost by BA & QF.
If you want to fly to a smaller city such as BNE, PER or ADL you will have to change at SIN, but then you already have to do so.