I've had the Public Betas installed through today (because they've been consistently more stable), and updated to my DEV Beta Profiles today so I can get WatchOS 3. I installed iOS 10 beta on my iPhone 6 without any issue (except it was slow to download)... but I got stuck trying to update the Watch. I continuously got "Watch OS 2.2.2 Installed - You're on the latest software".
Even though I had the proper profile installed, it seems the profiles all conflict, and clearing them out is a bit messy and I tried a bunch of steps to try to get it to show up... all which failed until the last one. For anyone else who gets stuck here, the only way I was able to get through this was to Remove ALL Profiles from the Apple Watch, then restart the Apple Watch & iPhone, then add the new profile & restart the Apple Watch & iPhone again. Then WatchOS 3 shows up for the software download.
I'm not sure if every iPhone restart is 100% necessary, but the Apple Watch restarts definitely were for me.
EDIT: Quick note... There was no need to downgrade or restore the watch or phone to any factory settings. I did however reset all network settings at one point. I don't think it helped, but if the above doesn't work you can give it a try too.