You can't just randomly cut part out of the tire.
It has to be measured and the percentage of contraction of the wooden wheel has to be taken into account. Too loose and it falls apart, too tight and it breaks the fellows. You use a measuring wheel to go around the wheel and transfer the measurement to the tire.
Same thing about heating the tire, too hot and you ruin the steel, not hot enough and it won't slip over the wheel. A second too long and you burn the fellows. Also you need to drive the tire over the wheel and get it centered on the fellows evenly. Then you need about four or five guys ready to dash water on the tire to cool it before it burns the wood too much.
White oak is fine for heavy wheels like those. It's the finer, lighter buggy wheels that require hickory. And the wood has to be rived from a green log, not sawmill lumber. Sawmill lumber will break on the diagonal grain line created by the saw. For good strong fellows the grain has to run perfectly straight along the piece.
Like I said, maybe not well enough, I know how to do and so, therefor I know enough NOT to do it.
Get an expert/professional.