This is extremely frustrating however the same would apply even if it were not a Motability vehicle, the only other possibility is that outside of Motability we could phone around to find an independent tyre fitter who would come out in the evening/night. The problem is more because there is no spare wheel. If it were me I would have ordered a taxi straight away (which I think you have done) and make sure a family member had the key for the car so it could be sorted while away, again not ideal but its a solution.
Fingers crossed you will be good to go now or very soon to get to airport (saying you would drive there with a flat, again while frustrating, is not going to work).
