Is Zakat obligatory for my mother on the money she lent to my father and brother, which they are repaying partially each month, and which is entirely spent on her medical treatment and essential household needs?

What the mother lent to your father and your brother is subject to the rulings of zakat on money lent to others.

If the debtor is able to repay, the mother should pay zakat on her money every year, and her spending her money or assisting her husband with it does not exempt her from zakat.

If the debtor is unable to repay, zakat is not due on it until she receives it. Then she should pay zakat for one year, or for all years upon receipt. If she receives it in installments, she should pay zakat on each installment upon its receipt.