An AONE6 originating in the UK is just about as expensive as it gets. Again, it's your business or money, but consider the following tweaks -
Code:
From To Top class/notes
GIB/IST LHR J
LHR LAX F
LAX LAS Not OWE; AA or car
LAS YVR Not OWE; AS Y/F or PR J
YVR JFK F
JFK DFW F
DFW YYZ J
YYZ MIA J
MIA JFK F
JFK GIG F
GIG SCL J
SCL AKL F
AKL SYD J (F if LA)
SYD DRW J
DRW PER J
PER NRT J
NRT HKG F
HKG BKK F
BKK HKG F
HKG JNB J
JNB LHR F
LHR GIB/IST J
If you went to Gibraltar to start the trip, you'd save something like £1,700 before taxes; if you started in Istanbul you'd save £2,400.
The above uses non-OWE flights for the open segments. LAS-YVR on AS (Alaska Airlines - BA miles but no tier points) is US$135 in Y; or for something unusual you can fly to Vancouver on Philippine Airlines (PR) in J for US$350 or so. Between LA and Vegas there are umpteen cheap flights, or you can skip airport hassle and rent a car - 5 hours and around $100.
I think it hits all your key stops. It cuts out a couple of plane changes (mainly ORD and DFW) but that's not a terrible thing IMO. You could also go NYC-YYZ and save the DFW flight; just go LGA-YYZ instead of JFK-YYZ; that way you can ride up front on an MD-80 rather than an Eaglet. Maybe you could use that coupon to add back SIN to your itinerary, e.g. AKL-MEL-PER-DRW-SYD-SIN-BKK-HKG etc.
You could also save an additional segment by going MIA-GIG in a 767 with quite good service. If it were me I'd cut out MIA-JFK and go MIA-GIG, then use that coupon for JNB-CPT-LHR at the end.
You can use your two European stopovers in London going and coming, to help with your timing if that's a concern.
Just some thoughts.