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I would suggest you get a claims handling company involved. Insurance companies hate them because they have to pay for them on your behalf and if you find a decent one they will get the best value for your car. They will also handle suing the tulip out of the idiot that caused this for your inconvenience your insurance company won't care.
My brother has private plates and recently wrote his car off (well, he says someone pulled out on him...). He said he had to get the plates off and de-registered from the car before the insurance went through or else he would lose the reg. Insurance co went bonkers because all of a sudden the number plate changed and they tried to invalidate the claim. It all got sorted in the end but it was a bit of a faff.
I talked to insurance about it first, used online DVLA which instantly returned back old plates, which thankfully I have not thrown out. That old plate was how it was registered originally with the same insurance company anyway, all should be good...