You can order 3 months before your current lease ends, so if the lease ends on 16th May your order window should have opened a month ago?
If your new car hasn’t arrived by lease end Motability will extend your lease. There’s no issue with that.
You can leave your old car with the new cars supplying dealer, even if it’s a different brand. You do not have to return it to the supplying dealership.
If you have your heart set on a particular car, and you’ve had a decent test drive in one (at least a half day unaccompanied) then it’s best to order now. APs may go up in April or, worse still, the car may be removed altogether. If you order now you will not have to pay any increased AP if it goes up and if the car is removed you can still have it. However, get the dealer to agree upfront that if the AP drops in April you’d like them to adjust the AP down on the Motability system. It takes them a couple of minutes and doesn’t affect what the dealership receives for the order, nor does it affect the actual order.