tachograph:
Winseer:
How do you get to “buy” insurance on a vehicle that is both SORN and has no tax?
You phone the insurance company and say “I want a quote for insurance” 
Seriously, there’s no problem insuring a vehicle that’s not taxed, I’ve never insured a vehicle that’s sorned but I can’t see a problem doing that either.
Winseer:
A car is smashed up in a no-fault incident enough to cause it to be an insurance write-off, but driveable. The police say it’s OK to drive upto the MOT/insurance expiry. The insurance company insists on you continuing to pay the premiums (or firm keeps what’s already been paid) whilst the claim goes through. You presumably cannot re-insure a car that’s been already written off by THEM, but what happens with another insurer?
If a car is written off by the insurance company but still drivable you will be required to take it to a DVSA station for examination, it’s called a VIC (Vehicle Identity Check), the insurance will continue to be valid whilst you’re waiting for the VIC examination and assuming it’s road worthy you can continue to drive the vehicle, after that you’re free to insure the vehicle as normal.
There are about four grades of VIC ranging from A to D, the most serious A means you can’t drive the vehicle until after the VID, the least serious which would be a road worthy vehicle that’s been written off because the repairs would cost more than the vehicles value, means you can continue to drive the vehicle until the VIC.
I know this because it happened to me when another driver ran into the back of me while I was stationary, the insurance company wrote the vehicle off so I had to take the vehicle to a DVSA station and twenty minutes latter the job was done 
Winseer:
What happens if a car is written off, and you can easily fix it…?
The insurance company will pay you the market value of the vehicle and after getting the VIC you can fix it yourself.
So, the bottom line is - a written off vehicle can still be driven, insured, taxed, MOTed without anything making this prohibitive, such as “increased insurance premiums” which one would think should not happen to the “victim” driver who’s vehicle has been hit, eg. by someone with no insurance, or someone with full insurance, but currently acting as getaway driver for a robbery, or any other criminal act DD/DR etc?
Does a stoved-in car automatically fail an MOT? Knocking dents out is expensive, continuing to drive it without doing more than superficial repairs is a lot cheaper, and still obtains a payout for “market value” of car, without actually having to surrender said car to the insurance company?
Looking at the old bangers I drive for instance, I would imagine that a stoved in backside will amount to “more damage costs than the value of the car to repair” easily. Ideally, I’d want to (a) repair it myself (b) get a payout without having to surrender the car (c) get uninsured losses without surrendering the car and (d) not lose my no claims bonus. Oh, did I say I want to keep everything, and lose nothing - it all being paid for by some third party who’s already been put 100% to blame by the police, even if not by the defending insurance company? 