Long ago at an International Horn Society convention, I tried out a selection of wooden mouthpieces. While I couldn’t imagine playing on one for an extended period, it sure did feel nice and supple on the lips.
The tone quality was decidedly … well … “woodwind-like.”
This makes me wonder if a wooden screw-rim would ever be possible or practical, but given my occasional acidic wit it probably wouldn’t last very long.
Master woodcarver Robert Jones has taken this one step further. While it is not playable as a musical instrument, it sure looks nice as a piece of handcrafted art. See his wooden French horn sculpture at Village Soup.