On the Apple Watch, the software update can be downloaded through the dedicated Apple Watch app on an iPhone by going to General --> Software update.
To install the update, the Apple Watch needs to have 50 percent battery, it must be on the Apple Watch charger, and it must be in range of the iPhone. watchOS 3 also requires an iPhone running iOS 10 to install.
On the Apple TV, installing the tvOS beta requires the Apple TV to be connected to a computer with a USB-C to USB-A cable, with the software downloaded and installed via iTunes or Apple Configurator. Once a beta profile has been installed on the device through iTunes, new beta updates will be available over the air.
It is not known what new features might be included in the watchOS 3.1 and tvOS 10.0.1 updates, but they are likely to include bug fixes for issues that have been discovered since the release of watchOS 3 and tvOS 10.
We'll update this post with any new features that are found in the two betas.