It would be wrong for me to give you specific advice on your situation as I’m not sure sorry however, the topic. The topic has been covered before but can’t think where but may have been @on the spectrum.
Normally Motability require you to visit the dealer twice, once to oder the car and second to fill in any remaining paperwork and to enter your PIN. It sounds as the as the dealer is as much to blame by not following Motability’s rules. As for the car being in your name isn’t exactly true as Motability is the owner and your just a registered keeper!