This will help
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.
And because other things have changed, this may be useful:
Change the watch face on your Apple Watch - Apple Support
watchOS 10.2 re-enables left/right watch face swiping
• Open the Settings app on your Apple Watch.
• Scroll down and choose “Clock”.
• Toggle on the “Swipe to Switch Face” setting.
And this
Closing open Apple Watch apps
Double-press on the Digital Crown -> Quickly scroll to the app you wish to close, and swipe left on the app, then tap the big Red X.
Being quick to scroll the apps and swipe left can be a pain, so disable the following:
Apple Watch -> Settings -> Accessibility -> Auto-Select Focused App -> Off