No I-T-I at JFK, so the process will be the same as for all connections whether onwards international or domestic.

At GIG, the bags should be tagged to LHR, but will be offloaded at JFK for OP to cart through Customs and then drop post-Customs before heading through security and then off to his departure gate. If by chance, GIG won't tag the bags to LHR, the only difference is that rather than dropping the bags post Customs, OP will need to wait for a minute or two to have an agent print a new tag.

Because the US does not have "transit" once OP is through Customs and has dropped off his bags, he is free to head into Manhattan or off to his departure gate as he chooses.