Watches might be suitable for reuse so can be offered on sites such as Freecycle/Freegle or sold if in good condition. There is a market for second hand watches if they are a brand name such as Rolex, Gucci etc. Otherwise these can be taken to a local Household Recycling Centre and put into the reuse container (in a box).
Or it could be repaired at your nearest Repair Café where volunteers will try and fix it for a small donation.
If the watch does not work you can still recycle it via Recycling for good Causes as they take broken jewellery and watches to raise funds for charities. Unfortunately watches cannot be put into your green recycling bin. Please put it in your black/grey bin as a last resort.