Just call and get them to switch (for free) your flight to a JFK-LHR one. There are seats available in CW for miles on every evening flight, latest at 2235. Then you have a nice easy transfer to LHR, arriving 1030, and a full day in NYC the day before.

The novelty of the CWLCY service is not worth the hassle of dragging youself 2 hours across London on arrival.