The lounge rules have been gone through on this forum many times before. Access should be granted but they may quibble,
The OP was asking about checkin and I think that is even more clear.
You are checking in your bags to put them onto a long haul J flight, you are just sending them on a Y flight for a bit to get there.
So when checking in your 'destination' is JFK, just via LHR.

If you actually check in at the airport desk, you may get a BP with fast track on it. I'm not sure you would if you printed it yourself, but you may as well try. If you have both BPs and show the J one as well I expect they will wave you through.