Is it obligatory for a sick and unconscious grandmother to make up the fast of Ramadan, especially since doctors have despaired of her recovery?

If your grandmother is unconscious and does not regain consciousness, she is not obligated to fast or offer expiation, because sanity is a condition for religious obligation. However, if she regains consciousness for a period during which she could fast, then she must feed a poor person for each day she was conscious, based on the Almighty's saying: "And upon those who can afford it, a ransom of feeding a poor person." If she is unable to feed, then the obligation is waived.