Flying SFO-JFK-LHR-BUD. Currently on board SFO-JFK flight landing at 4pm, connecting to BA116 (JFK-LHR). Got an email just before takeoff at SFO that BA866 (LHR-BUD) is cancelled. Logging in, it seems I am autorebooked JFK-MAD-BUD arriving two days late, so obviously not acceptable. AA ticketed. Question:

1) I presume I'll have to deal with BA. What's the quickest/easiest way to find an agent at JFK. Is there someone in the Chelsea lounge?
2) Now the harder part. What are my options? All the LHR-BUD flights seem full until the 24th. LH has some options. What do I need to do to convince BA to rebook me to LH? Only OW option seems to be LHR-MAD-BUD with overnight in MAD arriving half a day late.

Thanks for the help.