I know the word horologist, but that seems to be for actual clocks. I don't know if it's the correct word, or if there's a more specific one.
Disclaimer: If you mean someone who makes something other than the mechanical assembly which runs a clock, none of my answers will apply. All of what follows is about people who are involved with clock and watch mechanisms.
Horology: the study of timekeeping, is the overall category of engineering concerning measuring time. It is a specialized term, most native English speakers wouldn't use it in ordinary discourse. George is a horologist, a physicist who works with the United States atomic clock.
Clockmaker, Watchmaker (1): Since it is very unusual for an individual to manufacture entire clocks or watches, these terms are both widely used to describe people who repair clocks and watches. Irene is a watchmaker who leases space from the jewelry store to repair watches.
Clockmaker, Watchmaker (2): These refer collectively to the companies which manufacture timepieces. Rolex is a Swiss watchmaker which produces approximately 800,000 watches per year. The employees of such companies who do the actual work may also be referred to as watchmakers. Ruth is a watchmaker who assembles winding mechanisms at Rolex, S.A. in Geneva.