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 over 50 percent battery, it must be placed on the charger, and it has to be in range of the iPhone.
There were no major changes found in the first watchOS 6.1 beta, so it's not entirely clear what's included in the update, but we'll update this article should new features be found in the second beta. The update may be focused on bug fixes and under-the-hood performance improvements.
watchOS 6.1 does work with the Series 1 and Series 2 Apple Watch models, which were excluded from the watchOS 6 release.
Apple said that watchOS 6 would come to the older Apple Watch models at a later date, so it's possible watchOS 6.1 will be the first version of watchOS 6 to work with the Series 1 and Series 2 Apple Watches.