I did the pleb version of a schedule flight. Not to NY, but MAN to CDG supersonic over the Bay of Biscay in the early 90's. Best experience for me, was the take-off, such power! So glad I did it, it cost me circa £500 at the time for the flight, day out in Paris (didn't need that as I lived in Lux), and 737 back to Man (where I was visiting my parents just so I could do the Concorde flight). Got to visit the flight deck too.
The one I flew on was BOAB, which is now parked up at LHR.