Swiss turning is ideal for this, on a Swiss turning lathe the headstock moves rather than the tool, which improves accuracy.
One question - why did watchmaking industry started/thrived in those rather remote places up in the mountains? I get that not everybody wanted/could be in Geneva, but remoteness helps little when doing business, apart from driving costs down. You need very specifically skilled people to get to those remote places. Some small village(s) say near Yverdon les Bains would make more sense to me, in the past and now too. On the way back saw new big glass&concrete building of Vacheron Constantin literally in the middle of nowhere, in between 2 forests and small road.
https://medium.com/watchmakers-ch/the-history-of-the-swiss-w...