Greetee

Too early an enquiry results in forcing the greetee into a conversation, but waiting too long results in a verbal collision or in missing your target.

The greetee could have responded with a request for delay after part of the process (as in "oh hi, give me a minute," and then attending to another matter).

They greeted the others with some kind of rhythmical hand gestures ending with leaning one shoulder into the greetee.