USD 1800 additional for an AONE (and USD 2900 additional for DONE). If you go hope it is LHR-JFK as that is a much better flight than JFK-LHR. Here is the rule
FIRST CLASS PASSENGERS MAY TRAVEL ON BA CONCORDE FLIGHTS FOR AN ADDITIONAL CHARGE OF USD1800.00/GBP1080.00/AUD2850.00/CAD2780.00/HKD13950/FIM10000/ESP275000/IEP1380/EUR1695 PER FLIGHT SEGMENT. BUSINESS CLASS PASSENGERS MAY TRAVEL ON BA CONCORDE FLIGHTS FOR AN ADDITIONAL CHARGE USD2900.00/GBP1750.00/AUD4600.00/CAD4480.00/HKD22450/IM16000/ESP450000/IEP2250/EUR2745 PER FLIGHT SEGMENT