Free Traffic Legal Advice

Is the other vehicle's registered keeper liable for the damage and costs caused by their vehicle?

No, but the driver of that vehicle, or the insurer of that vehicle (if it was even insured) is. But without any details of that vehicle, you're at a bit of a dead-end.

It might be a bit glib to say it, but that's what comprehensive insurance is for :-(

Surely the towing company is under contract to the police, who should then chase you the rk, who could then chase the MIB for the unknown/uninsured hit-and-run driver.

Uhh ohh you are straying into FoTL territory here ... of course the police force will have a contract with a recovery company.

Yes the police have the duty to recover a vehicle under certain circumstances, see Road Traffic Regulation Act 1984.

Unfortunately, I have no idea what FoTL means.
The advice that I have received from an online solicitor is that once the vehicle was damaged it became in breach of the regulations (no longer road worthy) and that the police then had the right to remove it. I don't know whether they should have advised the registered keeper that it needed to be removed beforehand. When the dust settles, I'll make a complaint to the Chief Constable, as advised on the police website.
For now, the costs are down to me.
My only redress is through the Motor Insurers Bureau or the other vehicle's insurer.

To expand upon what our illustrious, overworked and underappreciated co-admin said, FMOTL is a delusional and dangerous movement that seeks to argue that their consent is required to be bound by what everyone understands to be well known criminal law provisions, etc.

In the instant case, the OP appears to have fallen foul of far less well known quasi-criminal/administrative provisions that effectively punish not only those that have not been convicted of anything, in many cases have done absolutely nothing wrong.

Unlike those who have chosen to cast aspersions at the OP...

As has been explained, there are statutory provisions variously empowering and/or requiring public authorities to seize vehicles and charge for their release. Such powers must be applied proportionately, otherwise they are likely unlawful both under public law in general and through the application of the Human Rights Act.

I very much appreciate the advice from everyone. I'm certainly not a follower of FoTL.

My situation is unlikely to be unique but what it comes down to is that one minute my car was fully legal and the next it was in breach of the regulations (I assume construction and use). At that point, did I commit an offence under which the police exercised their right to remove the vehicle?

If I have committed an offence why haven't the police advised me of this, as noted by cp8759?

At present, the attending police officer is refusing to provide me with the licence number of the vehicle that hit mine, citing the GDPR. The vehicle that hit mine was left abandoned and he is waiting to hear from the registered keeper if the vehicle was stolen, TWOC or otherwise. This puts me in limbo as I can't contact the other vehicle's insurer. In addition, the towing company refuse to provide me with their own photographs, also citing the GDPR,  so I have no records to present to the other party's insurers when I find out who they are.

In practical terms, the towing company won't let me recover my personal effects unless I attend in person with the log book but the log book is in the car, which they won't open because they don't have the door key. I want the log book so that I can SORN the vehicle or have it removed by a breaker. The towing company have said that they won't release the vehicle unless their charges are paid and the will want a further £75 plus VAT for using their forklift to load the car onto any removal vehicle.

cp8759 Many thanks.

I shall follow your suggestion. I needed an objective opinion and yours follows my thinking. I will also put a warning shot across the bow of the towing company that there has been no police suggestion of any offence and that what they do with the car will be at their risk; not that I want the thing again!

I'll update this thread when there is anything interesting to those in a similar dilema and thank you all.