Nisaa
No. A woman in nifas (postpartum bleeding) does not fast, in exactly the same way she does not fast during hayd. This is not a rule invented for childbirth. It follows the same principle behind why nifas follows hayd's rulings: the two states share the same essential nature, and the majority of scholars treat them alike for prayer and fasting.
The textual basis for treating postpartum bleeding as a recognised state in its own right is the hadith of Umm Salamah (Radi'Allahu Anha): "The Nufasa' (a woman in nifas) used to continue to be in childbed for forty days at the time of the Prophet (Salla Allahu 'Alayhi Wa Sallam)." (Narrated by Ahmad, Abu Dawud, Ibn Majah and al-Tirmidhi). The duty to make up the fast afterward rests on the same hadith used for hayd, the words of 'Aishah (Radi'Allahu Anha): "When we would have our menses during the lifetime of the Prophet (Salla Allahu 'Alayhi Wa Sallam), we were ordered to make up the days of fasting that we had missed but were not ordered to make up the prayers that we had missed." (Narrated by al-Bukhari), set out in full at do you make up fasts missed for your period.
If nifas overlaps Ramadan
If bleeding begins on or before a day of Ramadan, that day's fast is invalid from the moment bleeding starts, the same way a period breaks a fast mid-day. Every day of Ramadan spent in nifas is a day not fasted, and every one of those days is owed afterward, made up before the next Ramadan arrives unless illness, pregnancy or breastfeeding gives a valid reason to delay further.
If nifas ends partway through Ramadan
Once she becomes pure and makes ghusl, she resumes fasting from that point, the same way a woman whose hayd ends mid-day does. See period ends mid-day in Ramadan for the two things that carry over unchanged: the day itself still does not count as a fast if she was not pure since Fajr, and the same Shafi'i and Maliki position against the Hanafi and Hanbali one over whether she must still refrain from eating and drinking for the rest of that day.
If bleeding runs past the recognised maximum
Nifas is not open ended. As set out in how long nifas lasts, the predominant opinion, held by Ibn Taymiyyah and reported by Ibn Qudamah as the position of most scholars, caps it at forty days, while the Maliki and Shafi'i schools allow up to sixty. Once bleeding passes the ceiling her own madhab applies, it is reclassified as istihadah, irregular bleeding rather than nifas, and istihadah does not stop fasting. She resumes fasting immediately, taking the same precautions against soiling and performing wudu before each prayer that any woman with istihadah does.
A separate concession, not this one
A new mother may have a second, unrelated reason to delay her make up fasts even after nifas has ended: pregnancy or breastfeeding can themselves count as a valid excuse for delay. That is about her ability to fast once nifas is over, not about nifas itself, and the two should not be confused.