I had to reinstall the iOS beta profile on my iPhone today to download iOS 8.4 beta. When I went to the beta website to download to my phone, I received the following screen. There was an option to download the beta program for the Apple Watch. Clicking on that option takes you into the Watch app, where you can download the beta profile. There is no beta update available as of yet, but I think this is an indication of a public beta for Apple Watch software.