Once things are settled and people are healed, Ohanzee will check for a spare.
Ohanzee finds a spare tire for the Bulldog. Just one, though, which is better than zero but less than the two you need.
The speed limit along this stretch is 55kph. It is up to Chino whether you want to drive faster (to flee the crime scene), the speed limit (to make it look like nothing is wrong), or slower (which will preserve the life of the tires a bit).
You can stay on side roads all the way to Snowmass (taking Road 19 to 17 to 16), at which point you will rejoin Highway 82. The speed limit on the highway is 100kph.
Presuming you drive the speed limit, this is how long you have until you reach:
Woody Creek (pop. 250): 8 kilometers, 9 minutes
Snowmass (pop. 250): 19 kilometers, 21 minutes
Basalt (pop. 4,000): 26 kilometers, 25 minutes
Carbondale (pop. 6,500): 44 kilometers, 36 minutes
Defiance (a.k.a. Glenwood Springs, pop. 10,000): 63 kilometers, 47 minutes
There will be cold resistance tests as long as you're in the car. Katsina's window-blanket solution will reduce the frequency of them, and the blankets ill provide Insulation bonuses, but the cold is still a concern.