It would have to be MAN-LGW-KIN
The first leg will be with a 737.
The second leg will be with a 777-200.
From LHR, you would have to connect via Miami most likely. American fly from there to Kingston on a BA codeshare.
(BA 207 LHR-MIA, B744 -> BA 5195 MIA-KIN, B738)
AA will be flying MAN-MIA next winter. You could try that option.