Chiming in late, but I had an experience like that--except I immediately contacted my own insurance company. The other person ran a red light and there were witnesses. My insurance company told me that they'd get everything started, but cautioned me that if I were at fault, completely or partially, then I might have to reimburse them. They got me a rental car within an hour, towed my car off to a body shop within a day, and got an adjuster estimate a couple of days later.
I thought my car was totaled, but it was still worth enough that my insurance company chose to repair it. I had to pay the deductible, but my agent told me that, as the other person was at fault, I would be reimbursed for it as soon as the my company had gone through the subrogation process and gotten paid by the other insurance company.
One I got the car back, I thought it was pretty much over, until I got two pieces of mail on the same day. One from the other insurance company saying "we have determined that the person at fault was not covered at the time of the accident" and one from my company saying the same thing, plus "your uninsured motorist coverage has now kicked in. Here is a check for your deductible. By they way, are you sure you have no medical expenses as a result of this? (I didn't)"
Just recently I got a letter from a collection agency. They are required by law to notify me that my insurance company has hired them to collect directly from the person at fault, in case I am pursuing a lawsuit.
The damages to my car were over $14,000. I had the rental car for a month. There were towing costs. No doubt my insurance company has found a bunch of administrative costs to add on. No wonder it was worth it to hire a collection agency.
When I bought my new car, I decided to pay for top drawer insurance, and it turns out to have been a good choice. Just got a new premium notice. No increase.