To activate the Warm-up and Cool-down segment types, go to the timer screen on your iOS device (iPhone or iPad). Tap the Settings icon in the top right corner. From there you can turn Warm-up and Cool-down on and off with the switch below the slider. The chosen profile settings will apply to the Apple Watch as well.
Note: The Cool-down segment will not be able to activate if the "Open session" setting is turned on since open sessions let the Main segment continue until it is stopped.