First, check your potential insurer needs a valuation. It is possible that some may not realise they'll get 'book price' for their car if something happens and that they need a valuation to get the full price. Footman James, for example, will insure without a valuation (but do want a few photos), but will only pay 'book price' whatever that is. They need a valuation for an accurate insurance figure. I'd say that anyone has has a really good car must have a valuation if they want to be properly insured. Otherwise should something go wrong one is going to be stuck in a negotiation about what the car would have been worth before it was stolen, or before the refuse truck rolled over it. I can't see how an insurer can pay a good price if the car has gone or is unrecognisable.
I know of 3 places that can do a valuation. I used Roger Edwards in N London. V helpful chaps. Michael at Silver Arrows will also do it, as will the guys at Chelsea Cars in Wandsworth.
I think if I had your car, Reggie, I'd definitely get a valuation.