I think it was mainly up to the military controllers in Europe whether they give a direct routeing or not. I'm pretty sure that the flight was OAT most of the time, so most of the GAT controllers had no idea about the situation, maybe not even talking to the aircraft nor having the flight plan.
I'm pretty sure if there was an opportunity and the situation was clear (eg. telling atc "we're a hospital flight") there would have been chance to go direct...
Anyway, hats off to the USAF for this mission.