Assuming it is a paid ticket, in discount economy fare class (note that only the most expensive econ fares are treated as full economy - these are roughly as expensive as business class), assuming you have a change of flight number at LAX (or alternatively stop over at least one night before continuing to JFK), and assuming you do NOT change flight number or stopover at HKG/SIN/BKK between LHR and SYD, then we have the following earning.
Note flown mileage is approximate and may differ from that used by QF. For convenience I have excluded any status bonus 25% QF Silver, 50% QF Gold, 100% QF Platinum.
SYD-LAX 7502 flown miles = 45 sectors, 7502 points (11253 points incl bonus for QF Gold)
LAX-JFK 2472 flown miles = 25 sectors, 1236 points if on AA/2472 points if on QF (1854/3708 points respectively incl bonus for QF Gold)
JFK-LHR 3466 flown miles = 25 sectors, 1733 points (on either AA or BA) (2600 points incl bonus for QF Gold)
LHR-SYD 10556 flown miles = 60 sectors, 5278 points if on BA/10556 points if on QF (7917/15834 points respectively incl bonus for QF Gold)