All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhiwasallam, is His slave and Messenger.
The woman in question should have fasted the month that she missed because of breastfeeding before the following Ramadan came. Her negligence of this duty is a sin since she had no excuse for the delay. She has to turn to Allaah in repentance, feel remorse for having done so, and be determined not to do it again.
Now, she is required fast the days that she missed. She is also required to pay the expiation for having delayed fasting those days until the following Ramadan. The expiation is to feed a poor person for each day that she missed.
Finally, if she broke her fast out of concern for her child only, then the guardian of this baby is required to pay an additional expiation. However, if she broke her fast because she feared for her health or for both her health and that of the child, then only what we have mentioned above applies.
Allaah Knows best.