The Torah commands regarding a newborn baby boy, “U’ba’yom Ha’shemini Yimol Besar Orlato” – meaning, he must be circumcised on the eighth day. The Gemara infers from this verse that a Berit Mila is performed on the infant’s eighth day even if this occurs on Shabbat. The obligation of circumcision overrides the Shabbat prohibitions, and thus if a child’s eighth day falls on Shabbat, the Berit is performed on that day. This applies also if the eighth day falls on Yom Tob.
There are, however, several exceptions to this rule. One exception is the case of a baby born via Cesarean section, whose circumcision does not override Shabbat. Secondly, circumcision is performed on Shabbat only in cases of “Mila Bi’zmanah,” meaning, if the Berit takes place on the child’s eighth day. If the Berit is delayed, such as if the child is jaundiced or underweight, or for any other reason, it cannot be performed on Shabbat. If the physician declares the baby fit for circumcision on Shabbat, the Mila is delayed until Sunday, because a “Mila She’lo Bi’zmanah” – a circumcision performed after the eighth day – does not override Shabbat.
Moreover, in cases of a “Mila She’lo Bi’zmanah,” the circumcision may not be performed on Thursday or Friday, either. The Tashbatz (Rabbi Shimon Ben Semah Duran of Aligiers, 1361-1444) ruled that since the third day after circumcision is especially painful, and the child might require medical attention, a Berit should not be performed on Thursday in situations of “Mila She’lo Bi’zmanah.” Performing the circumcision on Thursday will in effect create a situation that will likely require Shabbat desecration, which should be avoided. The Poskim extend this ruling to performing a circumcision on Friday, as well. If the third day is especially painful, then on the second day, too, the child is likely to require medical attention, and a Berit should therefore not be performed on Friday in cases of “Mila She’lo Bi’zmanah.” This is the ruling of several Poskim, including the Taz, Kenesset Ha’gedola, and many others. (See Yabia Omer, vol. 5, Siman 23.)
Hence, if the physician declares the baby fit for Berit Mila on Wednesday night, the Mila is delayed until Sunday. Anytime a child does not undergo circumcision on the eighth day, his circumcision takes place on either Sunday, Monday, Tuesday or Wednesday, but not on any of the other days of the week.
This applies as well to the circumcision of a convert who must undergo Berit Mila as part of the conversion process, and to adults who were raised in remote regions and did not undergo Berit Mila as children. These situations of Berit Mila are considered “Mila She’lo Bi’zmanah,” and may be performed only on Sunday, Monday, Tuesday or Wednesday, but not on Thursday, Friday or Shabbat.
One exception to this rule is the situation of a child who is born during “Ben Ha’shemashot” (the period immediately after sunset) on Wednesday evening. For example, on a Wednesday when the sun set at 6pm, the boy was born at 6:10pm. Since it is uncertain whether to classify the period of “Ben Ha’shemashot” as daytime or nighttime, the boy in this case cannot undergo circumcision the next Wednesday, as that might be only the seventh day. The Berit is therefore delayed to Thursday. Although a Berit is generally not performed on Thursday after the child’s eighth day, this situation is an exception because we consider it a “Mila Bi’zmanah.” As Hacham Ovadia Yosef explains, the child in this case was Halachically (and not just medically) barred from undergoing Berit Mila on Wednesday, and thus Thursday is the first opportunity to have the Berit performed. It is therefore considered a “Mila Bi’zmanah” and may be performed on Thursday.
It must be emphasized that this discussion refers only to cases of “Mila She’lo Bi’zmanah.” As we mentioned at the outset, if a child is fit for circumcision on his eighth day, then he undergoes circumcision on that day regardless of what day it is, even if it is Thursday, Friday or Shabbat.
Summary: If a child is circumcised on the eighth day, the Berit Mila takes place on that day regardless of what day of the week it is, even on Shabbat. If, for whatever reason, a child was not circumcised on the eighth day, or in the case of a convert who undergoes Berit Mila as part of the conversion process, the Berit may be performed only on Sunday, Monday, Tuesday, or Wednesday. If a child was born during “Ben He’shemashot” on Wednesday afternoon, he undergoes circumcision the following week on Thursday.
Based on Rabbi Moshe Halevi's Menuhat Ahaba, chapter 1, pp. 22-23.