As Mamya indicates, a well-written rental agreement should make it clear who's responsible for looking after the garden of a property.
However (other than where there are shared garden areas or possibly where a local authority or housing association lets a property which is specifically intended for use by elderly or disabled people) I've never come across any agreement where the landlord has the responsibility. It's normally the tenant who has to maintain the garden.
For example, the tenancy agreement which applies to local authority housing in Sheffield states "You must keep any private garden or hedges to the property tidy":
https://www.sheffield.gov.uk/content/dam/sheffield/docs/housing/council-housing/DP19683%20You%20and%20Your%20Home.pdf
(Although that link apples to local authority housing, similar terms usually appear in private rental agreements).
If nothing is actually written into a rental agreement then it's possible that neither party has any responsibility for looking after the garden. (i.e. unless one or other party voluntarily keeps the garden in order it can simply be left to grow wild, with neither party having any right to complain about the situation).