You could start with
https://www.google.com/search?q=landlord+sued+by+tenant&sourceid=opera&ie=UTF-8&oe=UTF-8
Perhaps look at
https://www.landlordzone.co.uk/news/landlord-liable-injuries-tenant. While not exactly the same, it indicates that a landlord can be held liable if they fail to keep the property "reasonably safe". This is why, above, I mentioned always complying with current regulations even if changes don't apply retrospectively.
So, the stairs may well have met regs when originally built - but they don't now. The landlord should reasonably know that they don't meet current regs. So there is a strong case for arguing that without improvement, the property is not "reasonably" safe. It's simply not possible to say in advance which way a court would rule - as that article points out, it depends on circumstances, and I'd suggest the landlord is more likely to be held liable if a child is injured than if a reasonably competent (i.e. someone who is not disabled such as to need better protection, and is mentally competent in terms of being able to see common risks) adult is.
While clearly a sales pitch, this article deals specifically with stairs
https://www.bartletts.co.uk/can-i-claim-compensation-from-landlord-for-falling-down-the-stairs/#:~:text=Landlords' Responsibility for Safety of,Landlord and Tenant Act 1985.
And again, this one makes it clear that if a landlord ignores a risk they could reasonably eliminate or mitigate then they can be found liable.
https://www.arthuronline.co.uk/arth...rd-be-held-liable-for-a-tenants-injury-part-1