I learned how to machine parts in school the traditional way using a watchmaking lathe, but that type of work is typically reserved for people who specialize in vintage restoration work. I work on more modern watches where parts are readily available.

That said, the types of tooling you’d use for machining watch components is pretty similar to any other machine shop, albeit at a much smaller scale. We used mills, hand gravers, files, teeth-cutting wheels, drills, etc. and even did the heat treating and tempering ourselves. A huge, huge part of machining at that scale is honestly set up and measuring. I remember it taking us almost two hours to get the lathe and milling attachment set up for cutting wheels, and only about ten or twenty minutes to cut the teeth.