What are the prayers that my wife must make up, given that her postpartum bleeding stopped after 21 days from childbirth, and she did not perform ghusl until 3 days after it stopped, and that the labor pains began before Fajr and the birth took place after the Friday Adhan?

When the postpartum bleeding (nifas) stops, it becomes immediately obligatory for the woman in childbirth (nufasa') to perform ghusl (ritual bath) and pray. She is not permitted to delay, even if this occurs just one day after childbirth. This is because the ruling concerning her prayer and fasting is contingent upon the cessation and dryness of the blood, or the expiration of the maximum period of nifas.

Therefore, your wife must make up the prayers she missed after the bleeding stopped. If the "birthing waters" that descended before Fajr were pure water without blood, then this is not considered nifas, and she must make up the Fajr prayer, and the Dhuhr prayer if its time had entered before the onset of nifas blood. However, if the birthing waters were accompanied by blood, then it is considered nifas, and she is not required to make up the Fajr and Dhuhr prayers.