Breastfeeding has nothing to do with ablution and does not affect it in anyway.
As for the missing Fajr Prayer because of this, we referred the question to Dr. `Abd al-Fattah Idris, a professor of Jurisprudence in Al-Azhar University, who said,
The time of Fajr Prayer is more than the time of breastfeeding a child. It is usually about one hour and half and no child takes all this time in breastfeeding. She really needs to arrange the time she spends in breastfeeding him and leave time for Prayer. This cannot be an excuse for neglecting the Prayer until the time is out. So the mother have to take care to perform the Prayer in its due time. In case she missed it one time, she would have to make it up and seek Allah’s forgiveness.