Every day, BA flies about a dozen flights from London to Terminal 7 in New York's JFK airport. London Heathrow doesn't offer US pre-clearance, and based on lengthy discussions on flyertalk, won't be able to offer it any time soon either. London City doesn't have pre-clearance.
One quirk though is BA1. The BA 318s from London City can cross the atlantic on a full tank, but can't take off from LCY fully fueled, so stop off in Shannon in Ireland. BA3 rocks up in Shannon after the CBP staff have clocked off for the day, but BA1 normally makes it in time for passengers to undergo pre-clearance while the plane re-fuels. As such, passengers on BA1 normally arrive in the USA pre-cleared, don't need to go through immigration on arrival, and don't need to collect their luggage + clear customs on arrival.
My understanding is that most people on the "BA Babybus" LCY-JFK services travel hand luggage only, and also don't connect. People with checked luggage are therefore unusual on the service, and people connecting with luggage even more so.
For the odd unusual person flying LCY-JFK-(elsewhere), with checked luggage, on BA1, what's the luggage transfer situation in New York? Is through-tagged luggage automatically transferred? Or do you still need to collect your luggage on arrival into JFK (though in the domestic collections area), then re-check it for onwards travel?