Background: I'm on a PS3. I had Blackwall up for judging as well as the Duchess. I did Blackwall and it was fine, then I saved, then did the Duchess, and it was still fine. I decided I wanted to change something with my Duchess judging, (foolishly) reloaded from that last save. Now I'm getting that same bug as everyone here where I click "sit on throne", it takes forever to load, and then it's just everyone standing in front of the throne and I'm unable to sit in it again.
I tried: changing the throne, leaving Skyhold and coming back, reloading from a save from before I judged Blackwall and from before I got Judging Rainier from the war table and from before I went to see him in Val Royeaux, and multiple variations thereof.
What finally worked: loaded a save from before Blackwall left in the first place, judged the Duchess (which worked), quit the game (without saving), loaded the more recent save from before I went to Val Royeaux, and now it seems to be stable.
So...the fix could have been the successful judging in a much earlier save and/or restarting the game, but either way it seems to have worked! Hope this helps someone else.