In the British navy, local time was kept starting at noon, when the naval day started. At the noon sun observation (if weather permitted), the watch (the working schedule, not pocket watches) would be reset, and the timing of watches would restart again. Watches were four hours long, timed with a half-hour glass, with a bell being struck for every half hour (ex: one at 12:30, two at 1, so on until eight bells in the watch when the next watch would take over). The exception were dog-watches which were two hours long and used to reset the watches, so that each man would alternate overnight watches. If the glass wasn't turned for some reason (highly unlikely), ships officers often had pocket watches by the Napoleonic period.

The other comment in this thread about chronometers is wrong, unfortunately. Chronometers were an invention of the 1760s, not the early 1700s, and were used for navigation, not timekeeping. The chronometer would be set to a predetermined time (GMT in the British navy) and checked at noon to determine longitude. In any case they were expensive and not in wide use until the 1830s. I wrote more about longitude and chronometers here.