Setting time zone for Apple Watch

I traveled from US Central Time zone to Athens Greece. My iPhone automatically switched to the right time zone, my watch did not. My wife's and my daughter's watch automatically switched. All the advice on questions does not work. You can only adjust time by 59 minutes. If I use "language and region" settings on my iPhone's Apple Watch app, neither the "mirror my phone" nor the "custom" solves the issue. If I use Custom and try to select region, my iPhone just keeps telling me to turn on my apple watch, make sure it is nearby, and is not in airplane mode. None of those is the problem. How do I set my watch to be in Athens Greece time zone?