What kind of wood are these made of? A ring will get a lot of knocks and wear. Moisture rules out shellac and lacquer. I would think a spar urethane , but i do't know how thats going to feel on the skin. Will have to research more. I would want a finish to protect the base wood of the ring from wear. I don't have time now, but there may be an expoxy coating that would work here. Poly is the easiest to refinish, just sand light and recoat.
It also depends on what finish comes on it, are you getting it bare wood? If its stained you want to refinish the coat before you wear into the stain or it will stain uneven, unless you strip the whole wood piece and restain.
I'd have to investigate this more as i have not used it. I don't know how it would work on skin, I.E. sticking and slipping on and off a finger.https://www.amazon.com/Finish-Varnish-Provides-Gorgeous-Appearance/dp/B000PEGLBS
A ring i would think could be tough to do with self leveling polys, avoiding a drip mark on a round object. Perhaps even a wipe-on poly, but those are not thick so it would require several coats.
You may have to just experiment.