You could continue to fly BA on cash tickets, and credit to AA. AA only collects surcharges for award travel on BA and IB.

You should do a test award search on aa.com (no log-in required) to get an idea of what AA charges for Business Class between LHR and JFK. The "MileSAAver" redemption rate is 57.5K miles each way, but it can be challenging to find redemptions at that rate. (Is it your intention to fly Business Class on your family holiday trips?)

AA miles can be redeemed without surcharge on AY, CX, EY, JL, MH, QR, QF, and UL. AA awards include connecting flights at no additional charge (subject to routing rules), but stopovers (> 24 hours) would require two separate awards. Note that flying on a connection between LHR and Asia could significantly increase your travel time compared to a nonstop.

Good luck!