Ex student with mental health issues about to be homeless

What she probably needs most is someone to advocate for her and support her whilst she adapts and rebuilds her life.

Try these places:

www.nhs.uk/Services/Trusts/Overview/DefaultView.aspx?id=2709

www.mungos.org/our-services
whitechapel.online
www.hestia.org/ourjourney
www.mind.org.uk
www.elft.nhs.uk

There are many more in London but these are the Eastern ones.

Health services should already support and refer patients who are homeless or at risk of homelessness, as part of their safeguarding responsibilities and normal daily work. Many London hospitals provide a 'Pathway' service for homeless people.

Shelter has a free helpline and can signpost your ex-student to the help she needs: 0808 800 4444

She could try Shelter.
Can the hospital help at all. If it's a mental health hospital they may well have half way houses available. Then there's the council.
Try this for information

All hospitals have a "discharge co-ordinator" might be worth a try speaking to them.

Would the hospital social worker be able to help?

Is she still registered at uni? If so, she should be able to get some help from there, from student services or similar.

Other than that, I'm sorry, but I don't have any other ideas.

Can't offer any advice, sorry, but this situation has become so common that something needs to be done. Hostels are the answer, surely. They would give people somewhere to stay while they get their lives organised. I know there are short-term places for people to stay, but something more long-term is needed.
My daughter works with people in this situation, people who previously would have been in a mental health institution. Since they were closed, these people have been abandoned.