This is BE, so not helpful for Stravinsky, but I put it here for future reference. In the UK (in 2021, at least), the terminology is:
nursery - paid day-care for children 0-5 before they start primary school ("my daughter's in nursery three days a week"). Hours are usually 8ish to 6ish. The alternative to a nursery for working parents would be a childminder or a nanny. See dictionary entry above.
nursery school or school-based nursery - this is where the term 'nursery' gets complicated in BE. Nursery can also mean the youngest class of primary school for children aged 3-4. This is the class before compulsory schooling begins (the first compulsory class is Reception class, age 4-5. The following classes are called Year 1 (age 5-6), Year 2 (age 6-7) etc).
Sometimes these government-funded (eg free to the parents) nursery classes are in stand-alone schools (nursery schools), or they are just another class in the local primary (school-based nursery). Ordinarily these nursery places are part-time (9-12ish or 12-3ish), so some parents choose to keep their children in paid nursery places for the flexibility of hours. My daughter's nursery class (within a primary school) actually provides a full-school day in nursery (9-3) but this depends on the local education authority.
after school club (and breakfast club) - is usually provided within a school. You pay for it, and it'll be run either internally or by an external provider. My daughters' school has an after school club from 3:30-6 that you pay for. The school also run educational clubs too, for a nominal fee, such as sports' club, or art club, or choir, but these only last an hour and aren't aimed at parents who need childcare after the school day.