It is not a legal requirement to have a written tenancy agreement. However, just because you do not have a written tenancy agreement it does not mean you do not have a tenancy. As discussed here, if a person is living in property with exclusive possession and for a set term, they have a lease. Alternatively, if they do not have exclusive possession of the property they may still be a licensee.
In these situations, both the tenant and the landlord have certain rights and responsibilities, even in the absence of a formal written agreement. Significantly the landlord in this case is bound to perform their statutory duties of repair (s11 of the Landlord & Tenant Act 1985) and the tenant has certain protections under the Eviction Act 1977.
However, whilst you can still have a tenancy without a written agreement this will likely create several disputes that could have been easily resolved by reference to a signed agreement.