Doesn't third party insurance cover this? I can speak a little from experience on the other side of this coin. I once, had a kid(young p plater) pull out in front of me in busy, but slow moving traffic, who i hit in the side. We swapped details but seeing as he wasn't insured, i just took it to my insurance company who have a clause for when you have accidents involving uninsured drivers. They took all my details and said they'd take care of it. That's all the interaction i had they scheduled all the mechanics appointments and organized everything else. I never spoke to the guy personally throughout the whole process.
However, he did question the bill he was charged too, and rightly so. The mechanic bill ran just under 4thousand for a broken front right light and a pretty dented front right panel, and a whole lot of very very minor damage. Everything that was even remotely damaged was completely replaced and painted to 100% brand new quality. I was driving an old pulsar that was only worth about what they spent to fix it, and could have probably been fixed to good-enough standard for a tenth the price. Anyway, i felt like a bit of a dick for not calling the kid personally, but if you wait round on these things people realize they can deny that the accident ever happened and its very hard to later prover fault(its pretty hard to prove fault at all, but thats a different issue).
The sum of my story is, if the insurance company was responsible for organizing the fixing, you can expect that they spared absolutely no expense, and your probably sweet out of luck.