The Shulhan Aruch (Orah Haim 131:4) rules that when a Berit Mila takes place in a synagogue, the Vidui (confessional) prayer is omitted from that morning’s Shaharit service. Our custom is that all the Minyanim in the synagogue omit Vidui that morning, and not only the Minyan at which the Berit is taking place. Additionally, the Kaf Ha’haim (Rav Yaakob Haim Sofer, Baghdad-Israel, 1870-1939), in note 82, rules that Vidui is omitted at a Minyan on the day of a Berit if the father, the Mohel or the Sandak is present. Even if the Berit will not be taking place at that Minyan, Vidui is omitted if one of these three individuals is present. The presence of one of these three men does not exempt the other Minyanim in the synagogue from Vidui, but the Minyan in which they pray that morning is exempt. The original custom in Halab was that only the father, the Mohel and the Sandak themselves omit Vidui, and not the rest of the Minyan, but the accepted custom now follows the view of the Kaf Ha’haim.
The work Derech Eretz, which documents the ancient customs of Halab, mentions the practice of chanting Nakdishach on the day of a Berit Mila to the melody of the song "Ata Ahubi" traditionally sung at a Berit. It should be noted, incidentally, that the final stanza of this song, in common editions of the text, begins with the words "Hazek Eretz Toba." However, Hacham Nissim Zafrani testified that he saw in the old books in Halab the text, "Hazek Aram Soba," referring to the city of Halab. It seems that this was the original, authentic text of this stanza.
Summary: On a morning when a Berit Mila takes place in a synagogue, the Vidui (confessional) prayer is omitted from that morning’s Shaharit service in all Minyanim in that synagogue. Vidui is also omitted in the Minyan in which the father, the Mohel or the Sandak is present. In the final stanza of the song "Ata Ahubi" traditionally sung at a Berit, the correct text is "Hazek Aram Soba," as opposed to "Hazek Eretz Toba."