Meg
The vehicle is yours as validated by the finance company document and presumably the receipt details when you bought the car. Just make sure you get hold of the V5 registration document and send this to the DVLA to change registered keeper details. I don't understand the notice bit from the finance company, does this just confirm that the finance has been paid off.
Your ex partner may have made some payments, but I doubt these would be a much as the hire charges, you could have asked him for. Is he going to contribute towards the wear and tear to the vehicle.
Huckster