Is it obligatory for someone who broke their fast in Ramadan due to gestational diabetes and was unable to make up the days because of their son's continuous illness to fast the remaining days immediately before the next Ramadan, and is she considered sinful if the next Ramadan begins and she has not completed making up her missed fasts?

There is no blame in breaking the fast due to hardship, and the missed days must be made up before the next Ramadan. If one delays making them up due to a valid excuse, they should make them up after the next Ramadan without any additional penalty. If one delays making them up without a valid excuse until the next Ramadan begins, they must make them up afterwards, along with expiation (kaffarah) of feeding one poor person for each day. This is one mudd according to the Shafi'is and Malikis, and two mudds according to the Hanbalis (which is the more preponderant opinion). The expiation is binding upon one who knows its sanctity, and it is permissible to pay it before or after fasting.