There's an excellent train service between Darlington and London
Yes, but most people who'd fly to Heathrow wouldn't be going to London (i.e. central London); they'd be connecting to other flights from Heathrow.
One Northeast is a joke, it should be called One Tyne and Wear, or even just One Newcastle!
ONE has been abolished. As I understand it, there are now two Local Enterprise Partnerships covering what used to be the region of North East England: One covers Tees Valley (called the Tees Valley LEP), and the other covers Tyne & Wear and parts of SE Northumberland and some of County Durham (called the North Eastern LEP).
This reflects the government's regional agenda, i.e. to focus less on physical regions, and more on city regions and clusters of industries in particular sectors.
Newcastle Airport is therefore now free, if it wished, to market itself as the 'gateway to the Lake District' without stepping on the toes of the patch formerly covered by the NW regional development agency. Similarly, Teesside Airport could seek closer links with North Yorkshire.