I am flying out of Sky Harbor late this week on a QF numbered AA flight to LAX at 5.55pm. I have access to the Admirals Club (WP flying F) but wonder whether I can access "The Club" lounge which is a shared lounge during the day but becomes a "BA Exclusive" lounge from 3.30pm?
No. 3rd party lounges have different access rules for oneworld status.
For a lounge operated by a oneworld member, you can access the lounge with the applicable oneworld status or flight cabin, no matter which oneworld member you are flying with (other then the QR lounges in LHR and DOH or "additional" lounges such as arrivals lounges or lounges like BAs Concorde Room/Bar or QFs Chairmens Lounge).
For 3rd party/contract lounges, you can use either oneworld status or cabin class to enter the lounge, only when you are flying on the airline that contracts the lounge.
For example, at DFW, there is an AA club in each of the 4 terminals AA uses. There is also the shared contract lounge called "The Club at DFW". QF, QR, BA and JL contract the use of that shared lounge (with each carrier having their own room within the club). With oneworld status, you can access any of the AA lounges at DFW, but you can only use the JL Club room if you are flying on JL.
At PHX, there are 3 AA lounges which you'll be able to use with an F/J ticket or Sapphire/ Emerald status, however you can only access "The Club" if you are flying on BA (however, you wouldn't be able to access the AA lounges on a PHX-LAX flight if you had AA status).