In the event of an accident where the driver refused to provide their details (or failed to stop at all), you should contact the police, providing the driver’s registration, if possible, as they have powers to locate them.
However, if you do need to trace a driver who has failed to provide their details following an accident, the DVLA will likely honour your request for their details. You can request this information by submitting a V888 form to the DVLA.