Driving Through London, Car Shot, and Tires Do Not Blow
That part is plausible. Armored cars get armored tires. Or can get tires where if the rubber part is damaged, there is a rim that tire rides on. The car may not move as fast, but it is possible. This has been around since World War II. This type of tire was known as a combat tire.
See for example http://www.texasarmoring.com/armored_vehicle_bulletproofing_materials.html
Also see http://www.usarmymodels.com/ARTICLES/Tires/tires.html