The first place to start is the Vehicle Enquiry Service, which is a branch of the DVLA.
From here, you can find out information about the car itself (e.g. its make, colour, registration date, fuel type etc.), but not the owner’s details.
You may have some success tracking down the new owner through social media – or by enquiring through an owner’s club (if the car is a ‘classic car’). You can also try asking at the relevant dealership if you traded in your car.