The day nursery would require 'change of use' planning permission to operate in the first place but no further permission would be required for the use of the garden for play purposes.
All day nurseries must be on the Early Years Register, which is maintained by Ofsted. The maximum number of children on the premises is defined by the space available. There must be 3.5 square metres for every child under 2 years old, with 2.5 square metres for each 2-year-old child, and 2.3 square metres per child for those aged 3 to 5. There are also regulations relating to adult/child ratios, which also serve to limit the number of children present.
Chris