I have dowloaded the public beta profile but it still says that the latest version of watchOS is 2.2.1 - so watchOS 3 public beta is not offered for download. Has anyone else encountered this problem and is there any solution available?

It is most likely one of the following:
- you downloaded the iOS profile not the watch one to the watch (that's what I did - doh!)
- you didn't add the watch UDID to the developer portal
- you didn't upgrade your phone the iOS beta
In the watch app on phone check the installed profile and make sure it says watch beta profile. In my case I put the iOS beta on it instead.
Hope that helps.