Valve stem cap from a car tire ......... not a watch winder!
I completely agree. It's definitely a valve stem cap. No doubt at all.It seems like some posters won't use the experience and knowledge of other members regarding identification of finds , a quik internet search reveals that there are no antique watches from a Schroeder USA ! But numerous references to Schroeder industry's , and schrader Valves , screw it on a tire valve I'm sure it will fit! I don't think a watch manufacturer would bother putting there name on the winder knob.