[Zigûr]

I've done a quick search, and these are my apparent findings:

In The Hobbit, Gandalf mentions Bilbo's clock once, the narrator mentions Bilbo's clock twice, and Bilbo describes the interior of Erebor as a "clockless, timeless hole", which is the nearest we have to evidence that Dwarves did not have clocks. Of course other possibilities exist, such as that any clocks in Erebor had naturally become nonfunctional over many years. The word "o'clock" is used once on the sign put up by Grubb, Grubb and Burrowes announcing the auction of Bilbo's property.

Looking through The Lord of the Rings suggests that the word "clock" on its own occurs twice: once when Bilbo tries to put the envelope containing the Ring next to his, and once in Ithilien when Sam says that the time is "nigh on half past eight by Shire clocks, maybe." The word "o'clock" is used multiple times, but Gandalf is the only non-Hobbit to use it; he was familiar with Hobbits. The only non-Hobbit to whom the word is used is Mr. Butterbur, who evidently understands what the term means, although given the translation conceit we might imagine that the Hobbits were really saying "the Xth hour" when they say "X o'clock".

At one point when the hour is struck in Minas Tirith, however, Pippin muses to himself "Nine o'clock we'd call it in the Shire." This remark and Sam's about "Shire clocks", combined with Bilbo's complaint about Erebor being "clockless", suggest to me that Professor Tolkien imagined clocks to be an invention peculiar to the Shire and Hobbits, and perhaps only known outside it in neighbouring Bree-land, which was of course another place in which Hobbits lived in numbers.