The time on your Apple Watch will update when the iPhone that it is paired to updates its time. I've noticed when traveling by car into a new time zone that, even after crossing the time zone line for some distance (say 25 miles or so), my iPhone has not adjusted to the new time zone. I've seen this behavior again and again when crossing from Central to Eastern time zones driving across Indiana.
A quick flip of the airplane mode switch on and off will make the iPhone instantly adjust to the new time, assuming you're aware of where the time zone border is and when your iPhone should be changing time.
When flying, it's not as much of an issue as you turn off airplane mode at landing, which seems to prompt the iPhone to adjust the time as necessary.
It has always puzzled me why there is no system notification on the iPhone when the time adjusts due to a change in time zone. Even if you keep your phone in airplane mode but connect to a wifi network and the iPhone detects that it is in a different time zone on that wifi network, the iPhone will change time. And of course, not tell you that the time has been updated. Seems like a obvious thing to have a notification on a time zone change. I have not found an app to do this either.