You seem to be facing the same issue as in this post: Login seems to not be working inside Bitbucket SonarCloud settings
There as been some changes in the default way the browsers handle cookies. In the mean time if you are stuck because of this you can still manually add the
SameSite=None flag on the auth cookie in the browser debug tools to have the list of projects appear. It’s only temporary but might help while we deliver a fix.
But for new repositories you shouldn’t be using this UI to do the binding, you should just import your new repository from SonarCloud, and the binding is done automatically. This UI will probably be dropped sooner or later when we will add a possibility to bind an existing SonarCloud project to an existing Bitbucket repo.