You can see in your photos how the square block was put in the stone -- not a *perfect" fit, and secured with wooden wedges. That's the first step. Once it was the only step, and the block of wood extended far enough to become an axle (Roy Underhill has some photos in one of his books, and Eric Sloane has drawings in some of his -- I think in A Museum of Early American Tools)
Later the block was drilled for a metal axle, like yours.
The question is, is yours out of round, or off center. Sandstone wheels must not be left to stand in water. Notice in the article posted "Once sandstone is removed from the ground and "cured" (that is, its water or "sap" is dried out) it becomes impervious to water, salt..." If it gets soaked in one area, that area will become softer, and will wear more than the rest of the stone. As a result, the wheel will wear more rapidly than the rest of the stone, resulting in out of round. The only way out of this is dressing the stone back to round.
If it is off center, then it needs only to be properly centered. Not such a big deal. Put the wooden center in place and fix it there. After it is installed, then center and drill the axle hole properly. There are drill stands that can rest on top of the wheel and its center block to drill a perfectly perpendicular hole for the axle.
It will still need, probably, a bit of dressing to get it perfect, but nothing like the work necessary to get it out of round.
