Yes, using your digital watch is perfectly fine and Islamically acceptable. You don’t need to worry about a few seconds of variance.
Don’t let a few seconds cause you unnecessary stress. Your watch is more than accurate enough for all your religious obligations.
Islamic prayer times were originally based on physical observations (shadows and the horizon), which naturally carry a margin of error of several minutes. Shariat requires reasonable certainty, not microscopic atomic precision.A watch gaining or losing one second a week is exceptionally accurate. Even over several months, the drift would be less than a minute. Most prayer timetables already include a 1–2 minute “safety margin” to account for geographical and atmospheric variations.As you noted, even atomic-synced devices vary. This proves that searching for the “absolute” second is impractical. As long as you are confident the time has started according to a reliable source, your prayer is valid.To keep things simple, you can use the Islam360 app. It is highly reliable and provides accurate timings based on your specific location. You can sync your watch with the app once a month to account for that 1-second drift.