Okay, 2 different issues here:
Correct me if I am wrong here, but you are going LHR-SIN-SYD on SQ, then SYD-MEL-AKL on QF.
So, you will need to collect your bags and proceed thru customs/immigration etc in Sydney, then re-check-in at SYD Dom, at which points your bags will be checked thru to AKL. In MEL, you should not need to collect bags, but you yourself will have to proceed through immigration.
1) EVEN IF this was all on one booking or one alliance, you would still need to collect your bags and re-check in SYD, because you are entering Australia by virtue of doing a domestic flight (SYD-MEL)
2) If you instead had a booking of LHR-SYD on SQ, then an INTERNATIONAL connection SYD-AKL (without any domestic component) your options are
A) accept that the bags won't be interlined and allow enough time between flights to complete arrival proceedings, and then check-in again
B) beg at the time of check-in for them to be interlined, and you might be lucky, but know that this is NOT guaranteed and you might have to resort to A), which in some people's cases may have visa implications if not Aus/NZ citizens
C) Carry with carry-on only and approach the transfer desk for boarding pass once in SYD
D) rebook with a single alliance, or use a TA or consolidator (zuji/expedia etc) so it's all on one PNR (obviously can't be done for award flights)
Given you have the domestic leg SYD-MEL however, that's all moot as you need to pick up your bags and re-check regardless. I hope you've left enough time - sis and I missed a flight in HEL after slight delay in incoming flight, massive delay in bags (thanks FinnAir for the priority bag fail!) then a sprint to the checkin only to arrive a minute late and be denied checkin unless we wanted to leave suitcase behind. Last flight of the day too... :-(