Nisaa
No. A woman in nifas (postpartum bleeding) does not pray, in exactly the same way she does not pray during hayd. This follows the same principle behind why nifas follows hayd's rulings: the two states share the same essential nature, and the four madhabs treat prayer as suspended for both without exception.
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). As with hayd, missed prayers are not made up once she is pure again; only missed fasts are made up, the same asymmetry set out for hayd itself, where 'Aishah (Radi'Allahu Anha) said: "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).
On the day nifas ends
Prayer does not resume by itself; it stops for the whole of nifas and picks back up the moment she observes tuhr (purity), makes ghusl, and is able to pray. The same question that comes up at the end of hayd comes up at the end of nifas too, since nifas carries hayd's rulings on prayer without change: if purity arrives while a prayer's time is still open, is that prayer owed? The Shafi'i, Hanafi, and Hanbali schools hold that becoming pure with even enough time left for takbir al-ihram makes that prayer owed; the Maliki school ties it instead to having enough time left for five rak'ahs. The same disagreement over whether the earlier of a paired prayer, Dhuhr with Asr, Maghrib with Isha, is owed alongside it applies here exactly as it does at the end of hayd.
If bleeding runs past the recognised maximum
Nifas is not open ended. As set out in how long nifas lasts, the predominant opinion caps it at forty days, held by Ibn Taymiyyah and reported by Ibn Qudamah as the position of most scholars, 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 prayer. She performs ghusl, resumes praying immediately, and takes on the wudu-before-each-prayer routine any woman with istihadah follows, rather than waiting for the bleeding to stop entirely.
A new mother whose nifas coincides with Ramadan faces the same suspension for fasting; see can you fast during nifas for how that plays out, and why only the fast, not the prayer, is made up once nifas ends. The Mushaf is closed off the same way prayer is; see can you touch the Qur'an during nifas for that ruling and its exceptions. The mosque is closed off the same way; see can a woman in nifas enter or stay in a mosque for the passing-through and Eid exceptions that still reach her.