Right, here we go: if the car is uneconomical to repair, it's written off and the policy is cancelled immediately. It doesn't matter if it is unroadworthy or not. The vehicle only becomes their property once the claim has been settled.
The insurers will move the car once it's been written off, which will usually be before the claim has been settled. This way they prevent further claims being made for example if it's vandalised or another car hits it.
Hope this helps.