You wrote:
And in my Nexus I have setup the Public repository to have the 3rd party repository in the "Available Repositories"
The "Available Repositories" are the one that are available to be added to your group. In other words, it shows the repositories that are not part of your group. You should move the 3rd party repository to the "Ordered Group Repositories" to make it part of your group.
Not a very intuitive UI section imo.