Hello @Doughnut and welcome to the forum.
Dealers do not usually ask for the AP to be paid until the car is collected. Please note that most are also offering to take the “new car payment” (currently 750 GBP I believe) from the AP at the same time.
Dealers can offer un-registered stock that they may have available to Motability customers if they wish, but as they make more profit from private sales/lease, some insist that you need to place a factory order. You may stand a better chance of getting one at the end of the month as Motability cars do count towards their sales bonuses from the manufacturer.
Ring round a few dealers or visit in person, and see what they have to offer.
Good luck with your search.
Sean
Current car Hyundai Kona Ultimate EV