Unless you want to spend time dong int transit, via MEL/SYD/BNE/PER, the only way is to go with EK on the night flight from ADL - DXB - LHR, if you book on the QF website, you will earn QF points/QPoints, and QF SC as if flying on a QF aircraft.
Choose aisle seats, as EK don't have premY, but if you have aisle seats, at the least, you can stretch a bit into the aisle at night, as not too many people walking in and out, as most are asleep.
Or emergency row, for you and the son ahead, as this gives a bit of extra room, or son in exit row, and you in front.
I am also based in ADL, and have to transit MEL/SYD/BNE to get to NZ where I holiday a lot, either flying QF (less now), or VAd to VAi via BNE.
Look at the QF booking website anyway for fares.
Without any status, you will just have to pay the cash and earn QF points and QF SC, and use the points, maybe for something else, maybe hotel stays, or as we say on here, getting a toaster as a reward from the QF store.
If you do want premY, the best bet is to fly QFd Aust dom earlier on in the day, to get to PER to meet the 787 from there to LHR, at least you know you will have the same seat all the way.
If you fly MEL/SYD/BNE to SIN, and then on the SIN - LHR sector, you might find your preferred seat given to someone else on the SIN - LHR sector.
Edit: if you are able to get (on one booking), ADL - PER - LHR, one PNR, in PER you don't have to go to the check in in the terminal, as your passport is swiped by QF in ADL, and the check in staff in ADL would be/should be able to issue you BP all the way to LHR, and you just need to go via passport control after going from T4 PER -> T3 PER.
Or as I mentioned, go EK direct from ADL - DXB - LHR.
Or, for solely QF, you can also phone up, with long wait, and buy what they call Comfort seating booking, ie buying 3 seats for 2 people.
Sorry, I know I waffle a lot too much.