Gemara

(R. Yochanan): If one was Makdish a pregnant animal for a Chatas and it gave birth, he may bring either of them for his Chatas.

He teaches that one can (be Makdish a pregnant mother and) be Meshayer (stipulate that the fetus be Chulin or a different Korban), and that Ubar Lav Yerech Imo. (A fetus is considered an independent animal. It is not just a part of the mother.)

Yevamos 46b: Since R. Yosi holds that a convert must circumcise and immerse, Tevilah on Shabbos is Tikun (fixing him), and it is forbidden.

47a (Beraisa): If one comes to convert, we inform him of some Mitzvos, and their reward.

47b: If he still wants to convert, we circumcise him immediately.

When he heals, we immerse him immediately.

This is because we do not delay a Mitzvah.

78a (Rava): If a (pregnant) Nochris converted, her child need not immerse.

Since a fetus grows in his mother, she is not a Chatzitzah.

Kerisus 9a: Just like our ancestors required circumcision, immersion and Zerikas Dam to make a covenant with Hash-m (on Har Sinai), also converts.

Rishonim

Rambam (Hilchos Isurei Bi'ah 13:6): If a convert circumcised or immersed, but not both, he is not a convert until he immerses and circumcises. He must immerse in front of three judges. Since Beis Din is required, one may not immerse on Shabbos or Yom Tov or at night. If he did, he is a convert.

Kesef Mishneh: The Gemara forbids immersing on Shabbos because it is Tikun. The Rambam holds that one may not rely on this, since we permit immersing from Tum'ah, since it looks like he enters the water in order to cool off. Even though here Chachamim must oversee the Tevilah, it still looks like he cools off. Rather, the primary reason is because we do not judge on Shabbos, and conversion is called Mishpat.

Keren Orah (Yevamos 46b DH Tikunei): The Rambam said that we do not immerse converts on Shabbos because it is Din. The Kesef Mishneh said that even though the Gemara forbids due to Tikun, the Rambam rejected this, since we permit immersing from Tum'ah, since it looks like he cools off. Even without this, the Rambam does not hold that Tevilah looks like Tikun. He forbids immersing Kelim on Yom Tov lest one delay until then (but not due to Tikun). However, why didn't the Gemara forbid due to Din on Shabbos? I answer that we require laws of Din only for the act that enters him into the Bris (complete conversion). The Rambam permits immersing before Milah, so in such a case one may immerse on Shabbos. If we are concerned for Tikun, this is a Tikun, for without Tevilah, the Milah afterwards would not complete conversion. Really, it seems that Tevilah before Milah does not help, for it is like immersing while holding a Sheretz. However, the Rishonim say that it helps, and the Shulchan Aruch brings two opinions about this.

Rambam (7): If a pregnant woman immersed and converted, her child need not immerse.

Rambam (Hilchos Melachim 8:8): If a Yafes To'ar became pregnant from the first Bi'ah, the child is a convert. He is not considered the father's child in any way. Beis Din converts the child. Tamar was born from the first Bi'ah of a Yafes To'ar, but Avshalom was from after Nisu'in, therefore she was permitted to marry Amnom - "Daber Na El ha'Melech Ki Lo Yimna'eni Mimeka."

Or Some'ach: The Rambam was overly concise. If she immersed for conversion while pregnant, the child need not immerse. The Rambam discussed a Yafes To'ar who did not do so. Tamar's mother converted before giving birth, therefore Tamar was Avshalom's maternal sister. It seems that the Rambam was concise because he holds that the baby needs Milah (to be a convert), and he is not a convert until he circumcises. We do not say that since he cannot circumcise (now), he is like a female for whom Tevilah suffices. The Tevilah can precede the Milah.

Tosfos (47b DH Matbilin): 'We circumcise him immediately' connotes that it precedes Tevilah. We say 'he is not a convert until he circumcises and immerses.' However, we say that if a pregnant Nochris converted, her child need not immerse! The Ri says that there is different, for circumcision was not possible at the time.

Ramban (47b DH Nisrapa): Why do we circumcise first and wait for him to heal? We should immerse him first! Perhaps the Mitzvah is to circumcise first. Alternatively, since Milah is difficult, we circumcise first, so if he is hesitant, he will refrain. However, if he immersed before Milah, he is a convert. It does not matter which came first. A proof is that below we say that if a pregnant Nochris converted, her child need not immerse. If he is a male, the Tevilah preceded the Milah! If b'Di'eved it were invalid, the Gemara should have said that her daughter need not immerse.

Rashba (47b DH Nisrapa): The Ramban means that according to the opinion that Ubar Lav Yerech Imo and the fetus himself immersed, if you will say that another Tevilah is needed after Milah, Rava should have taught about a female. According to the opinion that Ubar Yerech Imo, there is no proof from there. Perhaps even if Ubar Lav Yerech Imo there is no proof, for sometimes the Gemara says 'Benah' and it really means 'her daughter.' I say that Tevilah must be last, for it is the primary conversion, and leaving Tum'as Nochri and entering Kedushas Yisrael. Even new Kelim of a Nochri require Tevilah for this reason. Regarding old Klei Nochri, one must Kasher them before Tevilah. There, perhaps Tevilah beforehand helps, since the Torah did not teach that the order is Me'akev. The Ra'avad says so. All the more so one, after separating from Orlah, which is like separating from Tum'as Mes (Pesachim 92a), one must immerse at the end. The Torah did not require Tevilah before and after Milah. However, perhaps this is only l'Chatchilah, but b'Di'eved the Tevilah helped.

Nimukei Yosef (Yevamos 16a, and Ritva 47b DH Shehuyei): Why do we circumcise first and wait for him to heal? We should immerse him first! I answer that Tevilah does not help before Milah, even b'Di'eved. It is like immersing while holding a Sheretz. We find that one who buys Kelim from a Nochri must Kasher them before immersing them. There, b'Di'eved if he immersed first it helped. Here is different, for the Tum'ah of Orlah is absolute Tum'ah. However, the Ramban said that b'Di'eved the Tevilah is valid. The Ro'oh rejected this proof. Inside the womb, there is no law to circumcise the fetus. It is like a female, for which Tevilah suffices. After the fetus is born, we circumcise him like we circumcise a Yisrael Arel. He is a convert even without this. However, whenever we require Milah at the time of conversion, if he immersed beforehand, he must immerse again after Milah.

Keren Orah (47b DH Matbilin): A fetus is not totally like a female, for after birth we may circumcise him. However, since now it is not possible, regarding this Tevilah he is like a female, so his mother's Tevilah helps for him, even according to the opinion that Ubar Lav Yerech Imo. The Ramban learned from here that Tevilah can precede Milah. If so, we must say that the Milah after birth is conversion. This is difficult, for in many places the Gemara says that if birth was in Kedushah, the child is a Yisrael immediately at birth. Pidyon ha'Ben applies to him (Bechoros 46a). What was Rava's Chidush? This Mishnah teaches that his mother's Tevilah helps for him! Rava teaches that all agree to this. It is not only according to the opinion that Ubar Yerech Imo.

Rashba (Megilah 20a DH v'Lo Tovlin): Just like our ancestors required circumcision, immersion and Zerikas Dam, also a convert needs these. Since it mentions circumcision before immersion, this teaches that if he immersed beforehand, it did not help, like it says in Yevamos. When he heals, we immerse him immediately, for we do not delay a Mitzvah. If it were possible to immerse before circumcision, we should do so, to avoid the need to wait until he heals!

Rashi (Yevamos 98a DH Nasa): If a boy was born Lo bi'Kedushah (before conversion) and born bi'Kedushah, if his mother later gave birth to a girl, it is his maternal sister and he is Chayav Kares (if he marries her).

Poskim

Rema (YD 268:1): If a convert immersed before Milah, b'Di'eved it is valid. Some say that it is invalid.

Bedek ha'Bayis: R. Yerucham says that the Milah must be before the Tevilah.

Shach (1): Therefore, he should immerse after Tevilah.

Gra (5): All agree that l'Chatchilah Milah must be first, like it says 'until he circumcises and immerses.'

Birkei Yosef (5): The Ro'oh rejected the Ramban's proof. Also the Rashba rejected it, and said that it is Me'akev to immerse after Milah, but at the end it seems that he was a little unsure. The Acharonim say that he should immerse afterwards.

Shulchan Aruch (269:4): If twin brothers were conceived Lo bi'Kedushah (before conversion) and born bi'Kedushah, they are liable for Eshes Ach (if one marries the other's sister).

Shach (6): The Beis Yosef and Bach say that this is even if they are not twins. This is difficult. The Mechaber connotes that it is only if they are twins! Also the Levush, Rambam and Magid Mishneh (Hilchos Isurei Bi'ah 14:14) connote like this.

Note: The Shulchan Aruch is from the Rambam, and the Magid Mishneh just says that the Rambam is from the Gemara (Avodah Zarah 57a).

Dagul me'Revavah: Rashi (Yevamos 98a) holds that it is even if they are not twins. Also the Rambam does not require that they are twins. It suffices that they are from the same mother, even if the first was conceived Lo bi'Kedushah and born bi'Kedushah, and the latter was conceived and born bi'Kedushah. A proof is that the Rambam in Hilchos Melachim says that Tamar was Avshalom's maternal sister.

Shulchan Aruch (124:5): If a Shifchah immersed in order to become a Shifchah, some say that her son does not disqualify wine (that he touches), even if his master transgressed and did not circumcise him, whether he is a child or an adult.

Shach (10): The Beis Yosef brought only Tashbatz, who says so. (Therefore, he said 'some say...') I say that this is clear from the Gemara (Avodah Zarah 57a), which says that if Bnei Shifchos circumcised but did not immerse, adults disqualify wine. Rashi (v'Chen) says that this is when the mother did not initially immerse, for if she did, they need not immerse. Also the Ran and R. Yerucham say so. If so, they are like Yisre'elim Arelim, who do not disqualify wine.

Rebuttal (R. Akiva Eiger): I say that the Mechaber said 'some say' because the Ramban holds that if a pregnant woman converted, her son requires Milah. Before this, he disqualifies wine. According to the Ro'oh and Tosfos, he is like a Yisrael even before Milah. In Siman 268 the Shulchan Aruch mentioned only Tevilah. Also the Acharonim did not mention Milah.