Is it permissible for a mother to atone for three years of accumulated Ramadan fasting by paying ten pounds a month, and is it permissible for her son to fast on her behalf?

If the reason for your mother's not fasting is an illness from which recovery is hoped, then she must make up the fasts after she recovers. However, if the illness is incurable or due to old age, then she must feed a poor person for each day. The amount for this is a "mudd" of food (750 grams of rice or similar) for each day, meaning 22.5 kilograms for the entire month, and monetary value is not sufficient.

The amount your mother pays monthly, if it exceeds one hundred pounds, is sufficient for expiation. What is due from past Ramadans is the expiation for three Ramadans (approximately 67.5 kilograms), which can be given all at once or in installments, and to one poor person or more. It is not permissible for the addressed person to fast on behalf of his mother.

If your mother is unable to pay the fidyah (expiation), there is nothing upon her at present, but the fidyah remains a debt upon her that she must pay when she is able.