I can't speak for the other countries you listed but in the UK we have the NHS which is free-to-use and at universities you can sign with a GP (doctor) who will make all important referrals, medications and stuff have different names over here, but obviously your doctor might know all about that. I don't know how bad your mental health is/can get but we have a hospital admissions process to the US if that's something which worries you except a court order is not required here for compulsory treatment, but the opinion of a psychiatrist, another MHA approved doctor and a social worker. But it's hard to get involuntarily committed here since there's few beds available and they're reserved for the most seriously psychotic and suicidal patients.

There's usually special counsellors and stuff at universities for helping International Students fit in aswell.

Hope you enjoy studying abroad if you choose to do so, it's a unique experience. x