It can be either:
"watch shop" - 363,000 hits on Google
"clock shop" - 181,000
Once we would have said the 'Clockmaker's Shop' - but they are now rare due to electronic and mass produced clocks and watches. These days when I need a watch/watch strap or repairs done I go to a Jewelers.
Agreed. For cheap watches and clocks, you go to a mass retailer or discount store. For nice watches or watch repairs, you go to a Jeweler. For nice clocks, you generally go to a nice furniture store.
There is a Clock Repair Shop in my area, but they are uncommon, and the gentleman that makes the repairs does this mainly as a hobby.