In my non-pilot opinion, it really doesn't sound like it was a HUGE deal just from the clip. Any idea of the actual separation at their closest?
From what I hear, tower was just trying to give Eagle an idea of what was around them, and when the Eagle started to go around, it sounded more like the controller was just worried what the planes were going to do.
Hazardous and nerve-wracking, yes, but not not an IMMEDIATE danger.