BA operates 744, A380, 777 from JFK.
The new new F is on the 789
Per BA site no A380 from JFK.
And this is what they say about the new F: "Over 80% of our long haul aircraft are fitted with the very latest First cabin and we are continuing to invest in the rest of our fleet. You can be certain to experience it on flights between London Heathrow and Lagos, New York JFK, Shanghai, Tokyo Haneda, Tokyo Narita, Hong Kong and Sao Paulo (subject to availability)."