I have always thought that, except for EK, the biggest competitors to QF is its own OW partners, especially given the SCs and QFF points earned on those airlines. This is especially the case for BA and CX, but also now MH and JL to lesser extents.
I disagree. Of the other 13 members now in oneworld, 7 don't fly to Australia (some not even anywhere close to Australia). LA, BA and JL only fly to one port in Aus. QR flies to 2. CX and MH fly to 6 each.
QF doesn't need to compete with QR, the link up with EK does that for them.
QF and JL compete on the NRT-SYD route, yet they codeshare elsewhere in Australia via SIN.
QF and LA compete on SYD-SCL, yet they also codeshare on each others flights.
QF competes with BA to SIN and LHR. QF has more AU-SIN flights then BA and more AU-UK flights. The issue here is cost between BA and QF. Most people won't go out of their way to get to SYD to fly with BA.
If anything, it's MH and CX that cause any issues for QF within oneworld. QF and MH could very easily have had a joint business much like the one they have with LA or AA, though QF and MH only really compete on the Kangaroo route.
The relationship between CX and QF is strained, but as for how much of that is on each side is up for debate. QF did used to codeshare on CX to FCO until the EK deal...
I see the biggest competitors to QF being non oneworld members. SQ, UA, NZ, EY and QF's own management.