Nouveau Musée de Bienne
Bienne's main history museum is not limited to watches, but it has a neat, permanent section dedicated to the city's horological past, with a few particularly interesting old Rolexes on display. It is located in a lovely area, a 10 min walk from the lake, a 5 min walk for the old town, and right next to a great antique clock store with a good selection of JLC Atmos. The museum has a pleasant garden where you can have a drink or ice cream when the weather is nice.
Further reading on Made in Bienne:
You can also listen to it below:
A 10 minute walk or a 5 minute bus ride from main train station. Here is the location on Google Maps. Open Tuesday to Sunday from 11 to 5. Up-to-date schedule on the museum website.
Estimated visit time: for the watch section only, no more than 1 hour, but there is more to see. A walk alongside the canal to the lake is recommended if the weather is nice.