What is the ruling on a father praying at home, given his suffering from psychological and physical illnesses that necessitate taking medication for his heart, diabetes, and blood pressure?

Illness is an excuse that permits abstaining from congregational prayer. The same applies to the fear of its onset, increase, or delayed recovery. If your father is harmed by going to the mosque due to his illness, or fears an increase in his illness or delayed recovery, then there is no sin upon him for praying individually at home. The illness that permits abandoning congregational prayer is one that makes going to the mosque difficult, not a minor ailment like a toothache or a slight headache. If your father is in such a state, then there is no harm in him abandoning congregational prayer. However, if he prays in congregation at home, it is good, and he will receive the reward of congregational prayer.