Halachic leniency does apply to nursing women on fast days, but such leniency is generally limited to the first two years of the child's life. If a woman has significant difficulties with fasting, a specific halachic question should be asked.
In a case such as yours, we would encourage you to attempt the fast, perhaps making arrangements in advance to help support you as you care for your son. If you do not feel well, you should contact a halachic authority during the fast to see if you can begin eating. This year, since the fast of Shivah Asar B'Tammuz is delayed from Shabbat to Sunday, there is greater room for leniency in cases in which someone feels ill.