I know nothing of making wagon wheels, nor of fitting new axe handles outside of what I just learned in this thread, so bear that in mind when reading my commentary here...
That being said, I think it highly unlikely that the amount the hole in an axe head grows when heated would have almost any impact whatsoever on this process. The coefficient of expansion is just so small, and the area it has to work on is so limited, I don’t think the numbers would add up.
By comparison, the amount the wooden handle would shrink and/or expand depending on moisture content would be quite large.
A wagon wheel or barrel hoop provides a great deal more room for expansion given their considerably larger size around the circumference.
Does anyone happen to know if there is more at play there than just the expansion of the hoop before placing it around the wooden frame? Perhaps the charring of the wood somehow serves a function?
@niteshooter, I’m not trying to call you out or anything here, kind of just thinking out loud. For some reason your post set some rusty gears to grinding in my head.