Control Center won’t appear on my Apple Watch

Use Control Center on Apple Watch explains how to access Apple Watch control center functions.

Open Control Center: 

Press the side button once to access the Control Center.

Use the buttons and screen on your Apple Watch explains "how to interact with your Apple Watch using the Digital Crown, side button, and gestures."

Supplemental information:

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.

If the flat oval side button is giving you difficulties, then look at

Use AssistiveTouch on Apple Watch - Apple Support

Apple Watch -> Settings -> Accessibility -> AssistiveTouch

can be used to access things like:

o Control Center

o Apple Pay

o Emergency Services and the Power Off icon (labeled "Hold Side Button")

o and many other things...