Except BA make lots of money on LHR-JFK which is the point. Moving two rotations out of LHR-JFK is not sensible as Delta and United would just up frequency and steal the passengers. The traffic would remain at LHR, it would not move to STN simply because BA moved two rotations to Essex. This is what BA found when they built up LGW. For every rotation moves out of LHR, too many premium travellers elected to fly LHR-XYZ with the competition rather than jolly down Sussex way. Harsh but true.
My point is how much profit is being sacrificed due to a lack of exposure in other markets, and how much of existing revenue can still be held on to if a move to STN is made? BA do make a lot of money on LHR-JFK, but healthy chunk of that is down to connecting traffic.
STN should be a cheaper option, so flaunt that fact. Marketing has changed considerably since BA at LGW was in it's prime, as have consumer attitudes. Low price finding websites didn't exist and are far more flexible than any travel agent or phone line could ever have been. That's why many of the average fliers fly Emirates and Qatar airways, because the comparison site says they are cheaper. If a notable number of premium passengers won't budge, then operate aircraft with a higher number of premium seats. Not such a problem on short haul routes as converter seating is used.
LHR is a bit of a pain in the butt - especially T3-T5 connections. Heck, even T5 connections are more troublesome than they should be and regulars know this. I'm sure there would be an incentive to cater to BA's wishes for a brilliant connections facility when STN is sold, as the new operator would be hungry for more flights.
How would Delta and United up the frequency if there are no slots for them to do so? Other than cutting into their own slots on other routes, which I don't think they'd want to do.
Maybe I am living in an idealised world. But I can't help but think what if? I realise things have happened in the past, but situations change and there there is more than one way of going about something. For it to have any chance of success it would take a hefty investment, which I suppose counts against the idea.