NFL has filed its motion for stay and sought expedited review. The players have committed to filing their opposition to the stay today at noon. (Not sure it has been filed yet.)
The NFL has asked for an "administrative stay" pending consideration of its application for a stay. These are usually routinely granted in emergency cases, but the players submitted a letter opposing it.
The options in the next day and a half are likely these:
1) The 8th Circuit will grant the administrative stay. This will be a decent signal that it finds something in the stay application worth thinking about. That stay will stay in place until the court rules on the application for a stay proper, at which point it will continue or dissolve the stay. If it dissolves the stay, it's pretty much game over for the NFL, at least in the short-term, unless they can get the U.S. Supreme Court to issue a stay. (A long-shot, although perhaps not in this case.)
2) The 8th Circuit will deny the administrative stay. This would be a good indication to the world that they aren't going to grant the stay.
3) The 8th Circuit will avoid the administrative stay completely and simply make a very fast ruling on the stay itself. I think this is unlikely but possible.
4) The 8th Circuit will do nothing. In this case, there is no stay and the NFL is at risk of contempt, and things will be uncertain. This is unlikely, but possible.
I think your expectation of mid-June is about right. However, we will probably get a very strong sense of where the 8th Circuit is well before that. If they grant a stay, it will be a strong indication they think Nelson erred.