There are direct train services from Newcastle to Manchester Airport.
There may be through trains from Newcastle to Manchester Airport, but they are not strictly direct, as they take a side trip into central Manchester on the way. A direct service would cut out Manchester and use instead the Stalybridge-Stockport line (a "ghost" line with, currently one train a week in one direction) and a link off the Stockport-Altrincham line into the airport. Absence of that last link means that an air service might just be viable.
I believe that there have been NCL-MAN services in the past, using sub-20 seaters, but I'm not sure if they lasted long. With today's increased flyer numbers and a few well-advertised codeshares Flybe might just be able to make a go of it.