No, your sensation follows the old wheel as it's shipped off to a recycling plant, where the rubber is stipped off, shredded, and turned into asphalt that gets used briefly in a road patch that is almost immediately replaced because the patch was bumpier than the road it was patching.

The metal of the wheel is melted down, then ends up in a youtube video where someone turns cursed metal (from the wagon wheel and also parts from the Pontiac Aztech driven by Walter White) into a bunch of sweet looking damascus knives, which feels like tooth with the enamel stripped off any time it touches something hot or cold for our poor wagon (who doesn't like to talk about what the video itself felt like, and swears he feels something every time someone watches it).