I don't quite understand the problem.
On the criminal side it doesn't matter who owns the vehicle, the police need to issue a request for information and Notice of Intended Prosecution against the keeper on the grounds of failure to stop and report, and driving without due care and attention.
If that is a hire company tthen they should then identify the hirer. If the hirer is a company then the fleet manager (or the person responsible for their vehicles) is legally bound to identify the driver. If they cannot they are guilty of not keeping proper records and the company will be fined under S172 of the Road Traffic Act. Because there will be no points the fine (against the company who fails to identify) is usually much higher.
In respect of the civil case (the accident) - which is entirely separate for legal purposes but which can be assisted by the criminal case - the damage to your vehicle indicates that the accident happened and the independent witness provides a track to the responsible party,
There must have been insurance on the vehicle whether it was a hire or not and hence that insurance policy is legally required to settle your damage and it does not matter who was driving, they still have to indemnify. In fact even if the car had been stolen, they still would have had to indemnify you.
The claim should therefore be directly against the third party insurer as I indicated several months ago (with information as to how we could assist you to resolve it).
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