A few more steps...
In order to install the watchOS developer beta, you will first need to enroll the iPhone paired to your Apple Watch in the iOS 15 developer beta and update to the latest developer beta build.
Please note Apple Watch cannot be restored to previously-released OS versions once the developer beta is installed.
Install the watchOS 8 Beta Software Profile
From the iPhone paired to your Apple Watch, go to BetaProfiles.com to download the configuration profile. Your iPhone will notify you when the profile has been downloaded. Go to Settings > General > Profiles & Device Management, tap the profile and then tap “Install” and follow the onscreen instructions.
Update your Apple Watch
Your Apple Watch will automatically receive the latest developer beta software if you have Automatic Updates turned on in the Software Update settings. To manually check for updates, go to the Apple Watch app on your paired iPhone, tap the My Watch tab, then go to General > Software Update.
You can always delete this installed profile anytime you want to stop receiving developer beta updates. All you need to do is go through the Delete Profile page, which has all the instructions for each of your devices.
However, Apple doesn't currently allow watchOS downgrades, so if you delete your watchOS beta profile, you'll no longer receive any other beta updates until Apple officially releases it.