Answer: For steel felloe wheels it is best to paint the metal parts the desired color before they are sent to the shop. This avoids one step in finishing them because you do not have to tape off the wood to paint the metal. This also prevents the paint from being drawn up into the wood spokes. It is OK to paint the hubs for wood felloe wheels. It is also OK to paint the rims if they are the clincher type.
Other types of rims or felloe bands are made hot to install them, so painting them is not advised.
There will be some touch up required on the metal, but it is not very much. I try to not damage the paint in the process.
For the final finish on a natural wheel it is best to use a marine or exterior varnish or urethane product. Be sure and use compatible finishes. Don't mix oil and water base finishes and stains. Prep the wood by lightly sanding it where necessary. Fill the pores with a paste wood filler that is used for hardwood floors. Stain the wood the color you want. Seal the wood with a sanding sealer. Sand lightly and apply the final finish. Spraying the finish works best if you can do it.
There are other ways to finish the wheels by using automotive clear coat or epoxy finishes. I do not have any experience with them so can't be of much help. A final note is to practice on some other wood first to see if your method works.