I've long said that the most effective way to link Heathrow and Gatwick by train is to combine the Heathrow Connect from Heathrow into London, which runs all the way into Paddington but duplicates the Heathrow Express in so doing, with the train from Brighton and Gatwick to Watford, which carries very few passengers north of Kensington, and which crosses the Connect route at Acton, where there is plenty of spare railway land for a link.
This would then provide trains from Heathrow to Clapham Junction and Croydon, and also from Gatwick to Ealing and Southall, as well as linking the two airports directly. None of these flows individually justifies their own train, but put them all together in the same train and you do so.