In most places in the UK, mental health support takes about 3 years to put you through the waiting list (Niche problems like for example trans healthcare are even worse, with the current estimated waiting list from approaching a doctor to receiving medical treatment from a GIC being 7+ years in some places of England), and I've know people who have had trouble with being dropped off the list when being transferred from the youth mental health support to the adult mental health support. Also, it still relies on you being in a fiscally and materially stable position where you can address these problems safely and from a place where you are not literally worrying about your ability to live to tomorrow, which for a lot of people who do not have active housing, is not viable.
Ah, I can't believe I managed to put this down wrong.
There are two required appointments from a GIC: First a psychiatric assessment from a therapist, Second an appointment with an endocrinologist. The timeline until the first appointment is approximately 3 - 7 years, and then you need to wait up to 2 years (At least, I've heard word of people waiting that long) for the endocrinology appointment, and then finally, you will receive treatment.
This happens even in countries with universal healthcare - the "needing money for a cab to get to the hospital" is a well known scam; the dedicated ones learn how to keep a wound open and bleeding constantly. (There was a notorious one near where I live, who managed to keep a bone exposed, allegedly - never seen him, so can't comment further on that).
In London, at least, my understanding is that most homelessness is a mental health issue; support is available, but not necessarily wanted.