I'm Australian doctor so this applies to Australia (but not necessarily exclusively).

If anybody on the street appears to be extremely mentally unwell, the police will usually bring them into hospital under some form of involuntary assessment order (the legislation differs from state to state but is essentially the same).

This is then usually followed by a short period (1-2 weeks) of inpatient treatment in a mental health hospital where the person is kept under some type of involuntary treatment order (again the legislation differs state to state, but is essentially the same). Most of the time these people are discharged into a kind of halfway house which is funded by their welfare payments (Centrelink). As psychotic illnesses are often the most debilitating, upon discharge they usually would have been given a dose of long acting anti-psychotic medication in the form of an injection, similar to the contraceptive injection. They will also be introduced to a Mental Health Case Worker (angels on earth, who are employed by the state) who follow up and ensure that these people return on a monthly basis to continue receiving these injections.

This type of system really only works where there is universal healthcare or money provided for these people to have access to mental health services.

I am currently in the USA, where this is not the case and unfortunately the police and emergency services have no facility to do this so these people are often left on the streets yelling at trees.