It's a hard problem, in part because you really need the state to address many of the problems with assuming fixed address. Instead of recognising the reality that people were people whether they had fixed address or not, the UK parliament decided to simply ban vagrancy [0]. Aspects of the act are still in force today.
> Is there no way to set up free post boxes with a fixed address for those that need them?
The currently governing party of the UK keeps parroting the line that "work is the best way out out of poverty" whilst passing laws and fostering policies that make that make life significantly harder for anybody homeless.
It's quite likely that the anti-"illegal"-immigration[1] "hostile environment" policy would view giving addresses to homeless people without proving their right to accommodation would result in the standard £10,000 fine per person the home office claims isn't here legally.
If an address was simply about receiving letters, then there is the Poste Restante service[2], which I only learnt of today. I suspect that few employers would accept that address and homeless people often don't have the proof of eligibility for work that they require.
What was once a universal healthcare system is actually no longer: In England, General Practitioners are required to write to patients and remove them from the register if they do not confirm receipt of the letter. This means being homeless without an address means emergency-only care, at least according to policy. [3]
[0] https://en.wikipedia.org/wiki/Vagrancy_Act_1824
[1] https://en.wikipedia.org/wiki/Poste_restante#United_Kingdom
[2] It actually affected a lot of people who were here legitimately. https://en.wikipedia.org/wiki/Windrush_scandal
[3] There are actually out of hours services that will see you without a fixed address. There may also be some local clinics that reach out to homeless people, but this is not by any central policy and if certain newspapers covered this appalling abuse of hard-working taxpayer's money being potentially used for homeless illegal immigrants then it would be swiftly dealt with.