As mentioned above look to see if there is award inventory in business class for your LHR-BLR flights. If so you can UuA(upgrade using Avios). Keep in mind that BA award inventory has dried up a ton (there's an almost weekly rant in the BA forums) but I've had good luck upgrading the LHR-India portions of the trip. So RDU-LHR(SWU)-BLR(UuA) is the best way to go.

No onward upgrades from Tokyo (and AA doesn't fly HKG) but it might be worth it to route to NRT on AA metal and slum up the short hop in Y if you can't find Avios upgrade availability. AA to PEK or PVG or ICN are also options. You will have to double connect though either in HKG or in DEL/BOM(or KUL) as no airline flies from AA's asia destinations to BLR.