>> It's probably because you have the NHS and socialized medicine, which would probably house the mentally ill homeless in some sort of mental hospital?

Unfortunately not. Mental illness is rampant among the homeless population. And, to be honest, even the most "normal" people will develop mental illness once forced to live on the streets for long enough.

Generally regarding the NHS and mental health, the subject is very controversial. Resources are continuously slashed and the care available is not adequate, in many parts of the country.