The Wheelwrights at Colonial Williamsburg make a wheel in a similar manner to what Leo described. I have had the pleasure of seeing some of their work during 3 visits, but never seen the complete process. They may have some information available.
"Like their Williamsburg predecessors, the wheelwrights who practice the trade at the Governor's Palace today start with a hub fashioned on a lathe from properly aged wood such as elm. A tapered reamer opens the center to receive a metal bearing; The wheelwright uses a chisel to create rectangular spoke holes around the circumference of the wheel. Carved from woods like ash, the spokes radiate to meet a rim of mortised wooden arches, called "fellies," that join to form a perfect circle."
From the Colonial Williamsburg website. Not sure what "properly aged" means.
The hub is (solid) elm for resistence to splitting. The blank is fully dried ... meaning 15 years or more. Wheelwrghts would put away blanks that their kids would be using. Spokes would be ash or oak typically, depending on size, with square tenons driven into the hand-chiseled mortices in the hub. The fellies around the rim are positioned with one for every two spokes (or you wouldn't be able to assemble them because you have to squeeze the spokes together to get them started in the mortices). Eighteenth century spokes had rectangular tenons, while newer styles used round tenons on the spokes. The fellies are doweled together (end-to-end) to keep them aligned when the tire is being shrunk on.
It is quite possible that they used Elm here in N America,(I'm not familiar with it) but the usual wood for hubs where White Oak and Beech in N Western Europe, there are quite a few complete wheels found in archaeological digs in the northern parts of Holland, Friesland and areas, as the people would use wheels with some broken spokes as foundation for their wells, those are dated back as long as 17 centuries.
Also the splitting of the hubs was not a problem especially later when iron bands/rings were used as these would prevent the splitting of the hubs, of course the weather in NW Europe is quite humid and wet, with wheels going through deep water-filled ruts rotting was a problem if softer woods like Elm or similar were used
Here's a picture from an old wheel with the iron sleeve in it and notice the banding, and lots of small splitting that the banding did prevent from growing bigger, grease would be used inside the sleeve to keep the wheel spinning freely.
also an archaeological write-up about the things found, like wagon wheels and canos (dugouts made from hollowed out Oak logs) sorry it is in Dutch maybe you can read it or translate it
Of course material choice and techniques will vary from one area to another, based on what woods are commonly available, tools, intended application, etc. Use whatever is available and appropriate in your area. Especially for decorative pieces, the wood choice is of little consequence. I agree with what Leo said too.