If they have re-rings then they are three ply. No question.Depending on vintage they would have had a dark mahogany outer ply. I doubt they were originally natural finish. That would be super rare. Without seeing pictures I think you have a set that was wrapped, then stripped then painted. It could be walnut or mahogany cortex finish which was counter top laminate. If so, you could strip away to your hearts content as you can't kill that finish.
There's definitely no wrap under the paint. Here are some pictures, see what you think. Note the blonde wood underneath (maple, right?) and red mystery stuff. Some of these are the nicks that it came with, some are what showed after some light sanding. Also, the BD hoops have no inlay grooves, which matches other natural finished kits I've seen.
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