I don’t think the start date means a lot in reality and I think Motability ask the dealer when they think your car will be delivered and set that as a theoretical date.
I know when I ordered my current car in mid October last year the lease contract said the payment would start in May 22 but around about March 22 I asked Motability to remove the car hoist from my order and they reissued the agreement and in that the payment were changed to start November 22 so those dates are pretty meaningless just like the delivery dates given for the car itself that I await delivery of.
Both of my agreement start dates BTW, May 22 and November 22 tie in with the date the dealer expected the car to be delivered.