What should I do regarding my wife's demand to move to a separate residence due to her inability to get along with my mother, knowing that my mother is ill and I cannot leave her to live alone?

Address the matter with wisdom and combine filial piety towards your mother with good companionship towards your wife. Inform her that her staying with your mother entails great reward, even if it is not obligatory for her. If she refuses, then provide her with independent housing. It suffices for a wife to have a room with its annexes. If that is not possible, then reside close to your mother to care for her and keep her company. Seek help from Allah and supplicate to Him.