Easiest plan is probably to contact the 3rd party insurer and deal with them directly as they have admitted liability.

They should bend over backwards to fix you up - avoids claim handler fees, they will hire a car directly and not pay high daily rates of claim management company, you can tell them the garage you want to use. They cannot write-off your car.

Based only on cost to repair. No allowance for car hire or any other expense or time

You are fortunate that 3rd-party's insurer has admitted blame. AIUI that insurer will study the used market and assess the cost of buying an identical vehicle, comparing that with their estimated repair, probably at one of their preferred repairers. If the car costs more to repair than replace, it will be written off and the insurer pays out and collects the damaged vehicle for salvage.

If you consider that a repair would be feasible, you have the option of keeping the car, and the insurer pays out its replacement value less their estimated salvage value (which they will have lost). I did that satisfactorily 6 years ago with my own car - still running.