Mi. you need to try and get the moisture content of the wood well below 20% 10 to 15% is optimum
for the tires if memory serves you will want to make the outside perimeter length the same as the perimeter of the wheel then heat the tire slowly and evenly until it has expanded enough to slide over the wheel then quench to shrink it to size. Failing this you would need to make a shrink-er which grabs the edges of the tire in 2 places then pulls the clamps together while you hammer the tire between the clamps. This is normally a process to temporarily tighten a loose tire.
to make the tires you just need a ring roller then weld the ends together and roll through until it is as near perfectly round as possible.
Hope this helps as I said it has been nearly 50 years since I helped make wagon wheels