If the hire car was from a separate company arranged by your insurance company (or a company they appoint) then you should be keeping the car until you have received the payment cheque (and you normally get a couple of days for the cheque to clear/find a new car etc).
If it’s from the garage the car is at then as soon as they deem the car a total loss their job is done.
If you want the car back act quickly. I knew ours was a write-of so wouldn’t let them take it to the garage and got them to come out and inspect at our house. Car would not run and has a bent shell.
When the insurance company write the car off the policy on that car ends. Your policy is then put on hold and you can switch the cover to a new car (paying any additional costs). If the car is safe and drivable and roadworthy then you may well be able to reinsure it (paying any additional costs). If it is cat C then it may need a VIC check but that is not a worry until the tax is about to expire IIRC. You may want to get a new MOT to check for anything dangerous.