The reality is that aircrafy generally just turn. If there was weather and traffic around as per your above description I would expect most controllers would have a little fat built into their plan expecting this. I would expect if an aircraft was told that their intended diversion was not available and they had to go then they would dial 7700 and go. But to answer the basic question if an Aircraft wont comply with an ATC clearance including doing something because they can't get a word in it is potentially very dangerous so would be an emergency and 7700 should be selected.