lawump

No run may score on a play in which the third out is recorded and that third out is (a) a force out or (b) against the batter runner before he reaches first base.
In your play, was R2 (the runner on second) forced to advance to third base?  The answer, of course, is yes.  The out against R2 at third base is a force out.  No run may score on a play in which the third out is recorded and that third out is a force out.
"Does the run score?"  Answer:  No.
"What should be the call?"  Answer:  The base umpire should say, "he's out".  The teams then move on to the 10th inning still tied. 
This is a Merkle's Boner play.  https://en.wikipedia.org/wiki/Merkle's_Boner

He's out.  The base umpire, by his signal and vocalization that the batter swung, is only officially recognizing what has already happened: the batter swung.  The batter did not swing when the umpire signaled.  He had already swung.  Because the catcher tagged him after he swung the batter is out.  There is no need for another tag and or throw to first.
In fact, what you described is very common in high levels of baseball.  College and pro catchers regularly tag the batter first and then ask the plate umpire "to check" with his partner.