Several of our customers (IT projects) are organisations for homeless. So I can tell you they do exist. Two types of referral exist: from the council, or a walk in from the street. There are also numerous TV shows that document the process. Sometimes it is called "sheltered housing" other times "homeless hostel" and other times "emergency accommodation". It is effectively all the same thing.

Like I said "living rough" i.e. the stereotypical homeless person is basically not a real thing in the UK. They do it because they either choose to do so or have other mental or social problems that prevent them from seeking free help.