In my NWA days, I took National Coach from LGW to London rather than the Gatwick Express train. It was a pretty ride -- rolling green countryside with sheep. And inexpensive. One bus route dumped you out at the Heathrow bus station from which you can take the "Connect" train over to Paddington and Bayswater or you could take the Picadilly Line through Hammersmith, Kensington, the "City," Islington, or clear over to Cockfosters if you had a mind to.
I don't have a mindset for Stansted, but I don't think connections are nearly as cheap or plentiful there.