The watchOS 4.2.2 beta, which was also seeded to Apple’s registered developers yesterday afternoon, is not available as part of the public beta program.
Members of Apple’s beta-testing program shall get their iOS 11.2.5 public beta over-the-air after installing a proper Apple configuration profile on their iPhone or iPad. As for the tvOS 11.2.5 public beta, you must first enroll your Apple TV at beta.apple.com.
Then, go to Settings → System → Software Updates and toggle on the Get Public Beta Updates option before you can download and install the latest tvOS beta software.