So

• LHR-SYD flying F or J
• SYD to PER by train
• Cruise from where to NZ (where). Then back to Oz?.
• PER (or NZ?) to SIN J
• SIN to UK flying F or J

QR fly's long haul into more AU airports than QF. For a time CX also did.
Flying AU domestic east Coast AU to/from PER can be expensive .4:15hr Better in business class
Trans Tasman AU<--->NZ are many airlines from/to many airports. More than just SYD-AKL

SYD is not the only international airport in Aussie.
Oneworld airlines to AU from Eu/Asia:- BA QR QF MH UL CX JL
And others in Star Alliance & Skyteam.
Going via USA can be more complicated

Wikipedia airport entries list airline-routes. Not always up to date or list (published) future routes
https://en.wikipedia.org/wiki/Perth_Airport
https://en.wikipedia.org/wiki/Sydney_Airport
https://en.wikipedia.org/wiki/Melbourne_Airport
https://en.wikipedia.org/wiki/Adelaide_Airport
https://en.wikipedia.org/wiki/Brisbane_Airport
https://en.wikipedia.org/wiki/Changi_Airport

Best of luck looking for the holy grail