Does the screen time include the apple watch?

Hello,

I edited this answer due to incorrect information I included. Thank you lobsterghost1 for your answer which made me realize my error. Even when selecting the "Share across devices" option an Apple Watch is not included in the device list when using Screen Time.

If the Apple Watch will be used by a child of yours the Family Group option may work better for you and have some Screen Time like capabilities.

Set up Apple Watch for a family member - Apple Support

"Turn on other features

In the screens that follow, choose whether to turn on and set up additional Apple Watch features. These include Location Services for the Find My app, Siri, Apple Cash Family, Messages in iCloud, Health Data, Emergency Contacts, Medical ID, Activity, Workout Route Tracking, and Photos."

"Set up shared contacts and Schooltime

Now you're asked to set up the contacts that will be available on the Apple Watch. To allow this, you need to have Contacts in iCloud enabled. Go to Settings > [your name] > iCloud on your iPhone and make sure that Contacts is turned on.*

Then you can choose trusted people from your own Contacts app to share on the Apple Watch. You can manage these shared contacts later, and set up restrictions, in Screen Time on your iPhone.

Finally, set a Screen Time passcode for the watch, and turn on Schooltime. When you're finished, tap OK to begin using the Apple Watch."

I hope this is helpful.

No. Screen Time is not a feature on Apple Watch.