Once the proper configuration profile has been installed from the Apple Developer Center, the new watchOS beta can be downloaded through the dedicated Apple Watch app on the iPhone by going to General --> Software update.
To install the update, the Apple Watch needs to have at least 50 percent battery, it must be placed on the charger, and it has to be in range of the iPhone.
It's not yet clear what features will be introduced in the watchOS 5.2.1 update, but as 5.x.x software, it's likely to focus on bug fixes and other under-the-hood performance improvements rather than significant feature updates.
No changes were found in the first three watchOS 5.2.1 betas, but should we find new features in the fourth watchOS 5.2.1 beta, we'll update this post with details.
Update: There appear to be issues with the fourth beta of watchOS 5.2.1 preventing users from installing it. Those who try are getting an error message reading "Not connected to Internet," so it may be best to hold off on installing watchOS 5.2.1 beta 4 until the issue is resolved.
Update 2: Several MacRumors readers have had success installing the update, suggesting the problem is now resolved.