Be aware people might have strong opinions about this and it could be a controversial thread, lol.
Some people like the patina of vintage pieces and wouldn't consider refinishing.
Just remember, it would be your set and you can do whatever you want with it. Secondly, I'm not a woodworker so I cant help much in this area but there are many YouTube videos.
I believe pieces are typically stained and then polished with wax or shellac. Again, this is personal preference. I would expect shellac to flake with age. The pieces in the picture may not have shellac and could have just been polished with wax.
If you are doing this for a high gloss finish as I mentioned in a previous post I didn't mean to suggest this is necessary. I was thinking about the scene from Netflix. Depending on the price of the pieces, you could get a reproduction instead that will come with a high gloss finish (and avoid all the work refinishing).
I like the set in the picture
Hi. I’m shopping for an original 1950s Mordovian (“Latvian”) chess set, and they all come with considerable wearing. What paints do I use to restore them to their original condition? And is there anything else I should know before I put a brush on an original set?