The insurance policy you have is irrelevant provided you have all the other persons info and they are holding insurance.
If they are uninsured then you'd likely need full comp with your insurance company then you just pay excess car gets fixed and they'll chase person responsible as long as you provide the relevant info 3rd party I think it'd be up to you to sort out as 3rd party is to cover if you hit another persons car, or your car suffers fire or theft
My smash repairer gave me the forms for claiming through a lawyer and as I said cost nothing so probably any repairer can point you in the right direction it's very common