No Tachanun - a so-called Perk of being a mohel

Anyone who grew up going to a shul where a mohel was in regular attendance recalls saying tachanun infrequently, in particular if the man was a busy mohel. Some mohels had a "siman" (indicator sign) that if he was wearing a tie in the morning, it meant he had a bris.

 We can well understand why if a bris is taking place right after davening, Tachanun, a prayer of supplication, would be omitted. Everyone feels the joy of the morning, everyone participates in the simcha, there is no room for a prayer such as tachanun.

 But sometimes the baby is not present, and the bris is not taking place immediately after davening, and sometimes it won't even take place in the same building, and in even some cases, noone from the family is present. So how could Tachanun be cancelled in these cases?

Please note that the definition of "Baal Bris" includes the baby's father, the sandak, and the mohel.

Here is my translation of the Shulchan Arukh and some of its commentaries on this subject.

Orach Chaim 131:4 
 “There is a practice not to fall on the face (e.g. say tachanun) in the synagogue on the day of a bris… Rama’s gloss: This is specifically if the bris will be in that synagogue. But if the bris will not be in that synagogue, even though it will take place in a different synagogue, then Tachanun is recited… Omitting Tachanun on the day of a bris is only relevant for Shacharis, because then [shortly afterwards] the baby is circumcised. But for Mincha, even when praying with/ in the presence of the newly-circumcised baby, tachanun would be said.”

 Ta”z (Turei Zahav)
Quoting his father in law, he notes that the rule about skipping Tachanun only applies to Shacharis, and not to Mincha. And, the Mincha which is required to have Tachanun is the mincha which takes place where the baby is not. However, if the baby is there, there is no tachanun at Mincha. This was also the view of R Shlomo Luria. It is the practice in Brisk, in Lithuania, and thus one should be lenient since tachanun is optional… This was also the custom in Krakow.

 Shaarei Teshuvah 131:10
Quotes the Eishel Avraham… that when the father is away, even if the circumcision will not be in the synagogue, since relatives are going to pray in the synagogue, there would not be tachanun. See also “Beer Yaakov” who says there are places where the custom is that when there’s a bris in the shul, where most of the city’s inhabitants pray, and it is the “elder” synagogue in town, then no other synagogue in town need say tachanun, as long as they are davening at the same time as the bris-minyan. Same is true for an individual davening alone at home

Mishneh Brurah 131 
21. Since the bal simcha was at davening, the simcha is shared with everyone and tachanun is cancelled.
22. Tachanun is not said in the synagogue on the day of the bris. Meaning, in the actual synagogue where the bris will take place, even if the baalei simcha aren’t davening there. The Acharonim wrote that if the baalei bris are davening there, i.e. the father, the sandak or the mohel [though not the kvatter], even if the bris takes place elsewhere, there is no tachanun in the minyan.
25. R Shlomo Luria, the BaCh and Taz all agree that when davening near the baby there is no tachanun. Eliyahu Raba decided that if they’re davening mincha in the baby’s house before the meal or DURING the meal, there is no tachanun. However, after bentching, they do say tachanun. The Chayei Adam wrote that this only applies when davening in the baby’s house. But once they’re davening somewhere else, even if they daven mincha BEFORE the meal and the Baal bris (father, sandak, mohel) is present, they should still say tachanun. Further, the baal bris himself (father, sandak, mohel) should not say tachanun at Mincha, even if it takes place after bentching, because it is his personal yomtov.

 Beer Hetev 
13 There is an implication that in the cold days (winter) whe brisses take place in the home, tachanun is said in shul. This requires further discussion. The “knesses hagdolah” wrote that the custom everywhere was that even if the bris was in the house tachanun wouild be skipped in shul if the baal bris davens there.

 If the child is born without a foreskin, and hatafat dam is performed, there is no tachanun