Your ex-partner still has a financial interest in the car, assuming that they paid the deposit and instalments to date.

The Registration document has no relevance to the legal ownership of the car, and, even if you could establish the identity of the Finance Company, taking over the payment of instalments does not necessarily mean that you can "adopt" the car as your own.

Are there any other joint interests of a financial nature? Are you the Insurance Policyholder, and, hence, know that the car is covered in that regard?

I really believe that in this instance you need to seek guidance from a competent solicitor.....or wait for some contact from the Finance Company and enjoy the car in the meantime.

You are the registered keeper, but not the owner. I don't think the finance company would be willing to disclose any information to you as you're not a party to the agreement.

I think the best thing you could do is contact the finance company and explain the situation to them. If your former partner keeps up the payments, lucky you. If he doesn't, it may be that they would be willing to make an agreement with you and end the agreement with him rather than try to repossess the car and sell it.

If it has less than a year left on the term, presumably there is a balloon payment due in October. Are you planning to pay it and keep the car? If not, I would just let events play out rather than try to do anything.