Now to be clear, the term "trench watch" was used in public advertisements at the time. A number of period adverts have been posted on this forum and others describing watches of this form as Trench Watches sold to military personal (officers were expected to buy their own), medical and public. No military provenance is needed. Indeed, "Trench Watch" was even a trademark of Ernst Borel granted in 1915 - I have one so marked as does Pilotswatch.There are 2 types of "trench watches".
Watches that were actually issued to military personnel and watches that were MADE to look like trench watches but didn't see "action" unless some guy really liked his watch and took it with him on the frontline or whatnot.
I can't tell whether it is a military watch or not... It seems kind of small... Does it have any extra markings ?