Just to confirm that BA will not transfer you or your luggage.

If you are alone then the National Express coach is a good option usually but frequency is very reduced currently so I'd recommend organising a prebooked car. If its on your dime then BA Transfer (no relation to BA!) is a cheaper option I've used many times, if I was charging it to my company card I'd use Addison Lee. Its around a 45min-1hr drive all being well on the motorway

BA seems to have moved a couple of routes that have few connections and high Y demand so are more appropriate for the Gatwick fleet - Islamabad is another