"Mental state" and "Mental health" are interchangeable, but "psychology" generally isn't. I think this is because it is more scientific than casual.

In the sense you are using it, "psychology" is similar to personality; it refers to a person's general mental disposition. In contrast, "mental state" and "mental health" are more temporary conditions. Mental state is the shortest term: "His mental state is poor. He is upset because his mother died recently." But that implies that he will eventually feel better. "Mental health" usually refers to a longer term phenomenon that might or might not change in the future, and probably was different in the past.

These three sentences are similar in meaning, but they do not sound equally natural or common. “My psychology is ruined” sounds unnatural. Native speakers do not usually use “psychology” to describe their own condition. “Psychology” is a field of study, not a personal state. “My mental state is ruined” is understandable, but it sounds dramatic or literary. It is grammatically correct, but not something people often say in daily life. “My mental health is ruined” is the most natural and common. It clearly means your emotional or mental well-being has been damaged. So in everyday English, “My mental health is ruined” is the best choice.