Is it obligatory to fast for someone who has partially recovered after an illness that prevented them from fasting, and is still underweight, while family members and the mosque's imam advise against fasting?

It is obligatory to consult the specialist doctor who supervised your treatment. If he confirms your ability to fast without affecting your health, then fast. However, if he believes that fasting will cause you health problems, then it is better to break the fast and make up for the missed days after Ramadan. Simply losing weight does not permit breaking the fast; rather, the criterion is the hardship, fatigue, or harm that results from fasting.