If you want to keep the car, do-not let it out of your possession. If it was recovered to a bodyshop or other garage, get it back to your own property ASAP. Keep it off road if it's no longer road legal. Why....
Somone rear-ended my miggy a few years back - the Aubergene Dream Machine. Some silly bint in a Frod Ka skidded into the back of me whilst I was waiting at some traffic lights. Anyway, next morning i dropped it off at the local Bodyshop. I fuilled in the insurance forms, and one of the questions was about disposal. I crossed that section out, and wrote in big bold letters that under NO circumstanced was the car to be removed from the premises without my permission.
A letter arrived on the Friday. Unfortunatley I was away abroad on the Friday, Saturday, and Sunday. When I got back Sunday night, I opened the letter, and it said to get round the bodyshop pronto to remove personal belongings as the car was being written off.
I went round Monday morning to drive the car away back to my home. At first I was told the car was 'round the back' - but I couldn't find it. Then they looked a bit more deeply into the 'puter, and Universal Salvage had arrived on Saturday morning and taken it away. I was Livid.
I kicked up and almighty stink, thretening to call the police because the car had been "Taken without owners Permission". I think I had Universal Salvage, the insurers and the bodyshop all dead to rights because i had copies of all the documentation. In the end I settled for (IIRC) £3000. Which I was happy with cos I'd only paid £2K for it in the first place - I was expecting a sub £1K offer.
Anyway - moral of the story - If you want to keep a write off, make sure you keep the vehicle on your own property, and under your own control.
Cheers
Malcolm