All the OW RTWs are booked in L as far as I am aware. However, if you take Marco Polo / AsiaMiles as your FF programme then you get 100% credit on CX, AY, LA, EI & AA, and 50% on BA & QF.

You could probably work out your itinerary avoiding BA and perhaps even QF, depending on what you want to do in Australia. On a well-planned LONE4 you ought to be able to get more or less to CX Gold (OW Sapphire, equivalent to BA Silver) doing something like this:
LHR-HEL-MAD-HEL-LHR-HKG-CMB-HKG-KHI-HKG-AKL-MEL-HKG-JFK-PHX-JFK-DFW-ANC-ORD-SFO-LHR
for over 60000 status miles (assuming you can find one of the ticketing agents who allows you to do the HKG transit from SYD to JFK); you need 60,000 for Gold on CX if you talk to them nicely (i.e. get them to waive the "going up through the tiers" rule).