The Control Center and its menu of buttons are supposed to be shown in every watch face on Apple Watches. Sometimes these interactions will not work when you try them, due to an undisclosed and intermittent bug. There are some possibilities in this article.
Many Apple Watch users have reported that they can't access notifications or the Control Center with a single touch on the watch face. The issue goes back to watchOS 8 and has been found to occur on different Apple Watches, so it is a recurring bug that spans hardware and software versions.
If Apple doesn't fix the watch face interaction problem, you can try one of the alternatives.
You can switch the watch face before attempting the other methods.
The control center and notifications interactions have been brought back by users who just swiped left or right to change the watch face.
It is not possible to restart an Apple Watch if it is charging, so undock it first.
Force restart your Apple Watch if rebooting didn't solve the problem. You should only use this as a last resort. Don't restart your Apple Watch if it's in the middle of an OS update.
Some people have been helped by unpairing the Apple watch. It's worth it if you don't see the gestures that reveal Control Center and notifications.
Your phone creates a backup of your Apple Watch before you erase it. The backup can be used to restore a new watch. You'll see the Start Pairing message after you unpair your Apple watch. It's possible to pair your watch to your phone again.
If you can't get any of the above suggestions to work, you'll have to take your Apple Watch to an Apple Store for repair or replacement.
If you solved the problem, you will have to wait for Apple to fix the problem in a future update. Hopefully we'll see a fix with the release of watchOS 8.7 or watchOS 9 in the fall of 2022.