A few years a go my local council paid in full for two tyres and a wheel no questions asked other than the pothole details and a photo of the damaged wheel/tyres. It was a huge pothole filled with water I was lucky that the tyres never popped but several cars were at the side of the road with flats. I also think it was no coincidence that a couple of years later the spring broke on the corner the wheel was replaced and the damper was leaking for both.
I have a claim going through just now for a couple of tyres, I had to replace both tyres on the axle as it's an AWD car which I doubt they will cover. They have asked a lot of details/photos/measurements and will only cover a percentage as the tyre was worn etc.
I got the contact details for the council via
www.potholes.co.uk, my council then passed the claim to 3rd party Gallagher Bassett to handle it.