There is no issue with adding your private number plate to your new contract hire or leased vehicle. AMG AUTOLEASE LTD recommend doing this procedure after delivery has occurred, as sometimes it may delay the delivery of the new vehicle.
After delivery and once you have paid your first monthly instalment you can then contact them directly to arrange for this to be done.
There may be an administration fee for them to do this and you would need to make sure the number plate is transferred from the vehicle prior to the vehicle being returned.