This will never happen. Even purely theoretically, it is extremely difficult to put forward a hypothesis about a factor (a set of factors) that would stop (or) the work of all calibers on the planet. Considering the introductory notes as purely fantastic, I would advise everyone who is not indifferent to watches to print out the manual "do-it-yourself gnomon" until the Internet sank into oblivion after the clock (post-apocalypse is so post-apocalypse).
Nothing special, they will be restored relatively quickly to within minutes. Humanity has long had clear records of time for each season, with a forecast for subsequent years
When sunset (or sunrise) comes, people will determine the time based on the day of the month and season, for example, today, 1 July 2019, we have (UTC + 3: 00) sunrise at 04:48 and sunset at 21:14. (Since June 22, sunset almost every day shifts by one minute to a lower hour, and dawn, on the contrary, to a greater hour, i.e. the night becomes longer). Dawn is counted from the direct appearance of the Sun on the horizon, it must be recorded on level ground, while sunset is considered the complete disappearance of the Sun behind the horizon.