As I experienced it at Agia Eirini: The chanting of the long Koinonikon by the Lampadarios during Communion of the clergy, pause NO MATTER WHERE IN THE KOINONIKON (the reason for this, as was told to me, was the "chalice waits for nothing". This is also reflected in that when the priest comes out for the Great Entrance they finish the Cherubic hymn by plain chanting the final words in the base of the mode), after priest says "With the Fear of God..." then a single chanter chants in 2nd mode very quickly, "Blessed be the name of the Lord, God is the Lord and hath revealed himself unto us." Then the Koinonikon is picked up where left off (sometimes, if stopped in an awkward spot, will move back to the closest martyria and go from there). If more time was needed at the finishing of the Koinonikon, then usually a Kratima was used. All the Koinonika I saw used at Agia Eirini had Kratima in them, so it wasn't an outside composition, just taken from the original composition itself. If the Communion line began to show signs of ending before the finishing of the Kratima, then the Lampadarios would just jump to the Alleluia.
So in this practice, before the Communion of the Laity.
As I observed it at the Patriarchate: Chanting of the Long Koinonikon before. I wasn't looking at the composition as it was chanted, so I cannot speak with as much authority, but it did seem like the Lampadarios had not finished the whole piece when the priest came out for "With the Fear of God...". After that, a single chanter chanted in 2nd Mode "Blessed be the Name..." After that, as Dimitri stated, they repeated quickly, and over and over, "Tou Deipnou Sou..."
One other thing worth noting is that at the Patriarchate, there was probably only 10-20 people receiving Communion, whereas at Agia Eirini the line was several hundred long and took, roughly, 10+ minutes to serve everyone. I imagine this has some effect on what type of practice a chanter must employ. As for myself, when I chanted at a larger church we used Full Koinonika and broke at wherever we were when the priest came out, but now that I chant mostly in mission parishes I use the "Tou Deipnou Sou" practice and a short or abbreviated Koinonikon during the Communion of the Clergy.
Another practice we used while I chanted at the larger parish was during the Communion of the Clergy, we chanted the whole psalm of "Praise the Lord..." (Psalm 148 and 149) in Heirmologic with an Alleluia after each verse. We did this in English using Papa Ephraim's compositions such as this one:
http://www.stanthonysmonastery.org/music/Orthros/b3170_Verses.pdf
We then added in the first two verses and added an Alleluia after each verse. We did this in the Mode of the week. After the priest said "With the fear of God..." we then chanted the Long Koinonikon in Greek. This we only did on Sundays where the Koinonikon was "Praise the Lord..." If another Koinonikon was in force, we just chanted the Long version of that.
To speak to your specific question, zinoviev, I have always heard "Blessed be the Name..." before the communion of the Laity, but chanted very quickly in 2nd Mode by a single chanter.