I damaged a Hertz car out of LAX a few years ago. I was surprised how quickly it was processed--not lightning quick, but I'm pretty sure I had a demand letter (IIRC from Viking Subrogation or something) within about a month or a month and a half after rental. I forwarded the letter on to American Express and heard nothing more about it.

At the location I worked, cars were taken to the body shop for estimates within 2-3 days at most, and we tended to have demand letters out to the customer within a week or two and never more than a month. (The cars were usually repaired within a week of the estimate, too, unless the body shop had to wait on parts.) I can't figure out why some companies take so long to get stuff processed--the sooner everything is done, the better our cashflow and the more likely we were to successfully get payment from the customer.