Apple Watch double click for approval disappears before any click can be made

The first (and simplest) thing to try if something isn’t working right is to restart your Apple Watch and its paired iPhone. If necessary (device not responding), try a forced restart. Both methods can be found here:

How to restart your Apple Watch - Apple Support

Restart your iPhone - Apple Support

If that doesn't help, the next steps would be to unpair and pair your watch again and restore from the backup just created. This should result in no loss of data. It deletes all temp and corrupted files and re-indexes the file system.

Unpair and erase your Apple Watch - Apple Support

Set up your Apple Watch - Apple Support

Getting the press and release timing just right can be an issue. You can make it more reliable by changing:

Apple Watch -> Settings -> Accessibility -> Side Button Click Speed

After changing this I never accidentally get the Emergency Services display.

“Side Button Click Speed” can adjust the press and release timing, so you get the Control Center display every time. Adjusting The “Side Button Click Speed” is also useful for Apple Pay’s double press and release. There is “Default” (sub second press and release timing needed), “Slow” (up to a second between press and release), and “Slowest” (up to 2 seconds between press and release). I like the “Slow” setting. And as I said, I do not accidentally get the wrong display.

NOTE: Slow and Slowest settings do not mean you have to wait a full second or 2 seconds between press and release. It just means you do not need to be super fast.