I think none of oneworld programs will take you to sapphire level after this journey. AAdvantage is one possible program to Gold (Ruby) if you take "Gold Challenge". It is also not possible to credit all of flights into one FFP. So probably QF Frequent Flyer for QF and SK flights?

(AAdvantage)
credit(actual flown)
BA: LHR-OSL 188(750)
BA: *OSL-AAL 125(219)
BA: CPH-LHR 153(610)

Total: 466 miles

AA: LHR-BOS 3265
AA: BOS-YYZ 500(446)
AA: YYZ-ORD 500(436)
AA: ORD-PDX 1739
AS: PDX-SEA 500(129)
AS: SEA-SFO 679

Total: 7183 miles

Grand Total: 7649 miles

*Not sure if this flight is eligible for AAdvantage because of not operated by BA.

(QF Frequent Flyer)
QF: SYD-LHR
SK: **AAL-CPH
QF: SFO-LAX-SYD

**Not sure if AAL-CPH is eligible route for QF FF.


If you don't care elite status, you can earn Asia Miles except SK flight.
(AA-100%(L), BA-50%(L), QF-50%(L), AS-100%(H))