I backed into a guard rail post in rural California in a Hertz car rented from LAX while trying to make a U-turn. The right rear quarter panel was dinged in pretty badly.

I drove the car back to LAX for my normally-scheduled return the following day.

I forget how long it was until I got the bill (this was around 2013), but it wasn't super short nor unreasonably long--I'm thinking somewhere around 6 weeks or so. It was a letter mailed by regular first-class mail from Viking Subrogation, a contractor Hertz uses to collect on damages. The bill was for approximately $600, surprisingly much less than I was expecting given the nature of the damage. I believe there was an estimate and some other supporting documentation enclosed. I can't recall if they had added any loss-of-use fees.

I had rented with my American Express Platinum with the Primary Car Rental Protection plan (primary coverage for $24.95 per rental). I emailed a scan of the letter and enclosures to American Express (I had already contacted American Express to open the claim and provided them with my receipt). I heard nothing more from Hertz or AmEx other than a note several weeks later that AmEx had paid my bill to Hertz's satisfaction. (I've since had another, much more minor, damage experience--a flat tire--on a rental I used a different card on, and that card has been less pleasant to deal with, demanding all kinds of documentation that Hertz wasn't willing to provide me. I never did get paid out on that claim by my credit card, but the dollar amount was low enough I just paid it out of pocket.)

Maybe there are some tidbits in that narrative you can pull out for your case, but here's what I think is worth noting: presuming that your credit card does not provide any type of rental coverage, you'll likely receive a similar demand letter and invoice as I did. You'll then have to either mail them a check (I expect they wouldn't accept a cheque drawn in UK funds, but some type of Western Union money order could work) or call them to provide credit card information. One thing they won't do is automatically charge your card: that's immediate grounds for them to lose a chargeback if you dispute the charge. (Rental car companies are not authorized by the Visa/MasterCard acceptance rules to charge for damages without the separate and express approval of the cardholder.)