Depends on the terrain, really. I can't imagine riding a bike with no tyres 10 minutes along a rocky path, let alone 10 miles...

Lets be charitable, shall we? They'd get pretty inventive, faced with 10 miles of walking taking a day, or on a bike taking an hour. Perhaps make hard rubber tires? Ones woven out of wire? Wood?

Bikes remain useful, even if they aren't up to modern road-bike standards. So useful, I think we'll see them for decades with ever more creative solutions to the problem of Bike Shop Closed, Oh What Will I Do?

I think walking slowly, carefully, and quietly would be much more survivable than tearing around in vehicles.

Well, its not all Mad Max. In a stable post-warlord community bikes would rule for a century. Tiny metal investment, fuel efficient, fast and adaptable.