I've made a couple rings but mostly use the lathe for clean up, sanding and finish.
When drilled from a single piece of wood you will get short grain on two sides making the ring more brittle. I laminate at least two mostly three thin pieces together, and oppose their grain, drill them out on the DP. Hole saw half way through, then forstner all the way through on the inside, hole saw all the way through to finish. I then sand the inside to fit and size.
I made a make shift poor example of a mandrel for the lathe tapered somewhat like a ring sizer. (I need to make a new one since I did that before I had a lathe at all, used it for the DP). I push the ring on the mandrel and finish with chisel and sanding.
I finished with CA. I tried carnuba wax but it wore off to quickly. The problem I have had is finishing the inside. I saw somewhere how a guy made a hole in a piece of wood attached to a face plate or chuck, pushed the ring in and then finished the inside. I haven't tried it yet, since mine are different sizes, plus I don't have a face plate or chuck yet.
Hope that helps, my wife loved hers. All my buddies who have come over to make one for their ladies have met with rave reviews as well. I threw together a simple wooden gift box to class it up a little more. My wife almost likes it as much as the ring. Simple top slides off or spins on rare earth magnets. Good luck.