The story as told here doesn't make a lot of sense to me in the UK (I assume it is the UK, from the reference to MOT and presumably the DVLA website at Gov.uk).
If the car was involved in an accident and the police were called to the scene, the police would have called a towing company to clear the road and put the car in safe storage somewhere, if in their opinion it was not driveable. That would not have had anything to do with the insurance company. Even if the police prosecuted the driver for being uninsured, they would still get the road cleared (and eventually recover their costs through the court).
At that point, the car had been assessed (by the police) as not driveable, but that is not necessarily the same as "a write-off". A police officer may well have given his/her personal opinion that it was a write-off, but that opinion doesn't have any legal status. The choice of whether to repair or scrap it would have been a matter for the insurance company, not the police.
The registered keeper (most likely the same person as the owner, for a private car) should have immediately applied for refund of the remaining road tax (if any) and a made a SORN declaration (Statutory Off-Road Notification) to the DVLA. After doing that, the car is no longer taxed, but the registered keeper is unchanged.
If the insurance company assessor decided the car was a write-off, at that point the insurance company should have instructed the owner to tell DVLA it was to be scrapped, or asked for the owner's registration document to do the paperwork themselves.
From the OP's post, it is not clear whether the owner ever made an insurance claim. A possible scenario is that the owner never submitted any documentation to anybody, so legally the car was never scrapped.
The post doesn't tell what the DVLA check said about the current registered keeper, but it is possible that the registered keeper is still the OP's friend. In that case, things need sorting out urgently, since the friend may be legally liable for things out of her control (and being liable for paying someone else's speeding and parking fines is the least of her worries there!)
I would suggest that since the police were originally involved in the accident, the best course of action would be to go back to them and take things from there. If the car is actually being used illegally, the police will certainly be interested in that fact.