If you have the car on Hire Purchase, you won't own the car until either the final payment has been made, or the Hire Purchase agreement settled early. The finance company retain title to the vehicle during the period of the agreement. Effectively you are renting the car from the finance company until the agreement is completed. Legally as they retain title to the car, then they should be informed of any change to the registration number. I suppose if you don't notify them of any change, they could view the change as possible fraud - ie making changes to the identity of the car.