Sounds like your watch is failing to start. Of the steps outlined there, my suggestion would be to first try a forced-restart by pressing and holding down both Digital Crown and side button for 10 seconds, until you see the Apple logo.
The restart can take a bit, particularly when you're waiting for it (like a watched pot trying to boil!), but be patient; you should feel vibration when it does restart and other tell-take signs that it's on.
If your watch still does not start, contact Apple Support, make the Genius Bar reservation or visit an Apple Authorized Service Provider. Could be a dead battery or some other hardware issue. What model watch do you have? How long have you had it? Anyway Contact - Official Support and find them https://locate.apple.com/
Good luck!
-- JDee