I am not quite sure what you men by older kids in your context. If they are old enough to go to school, they definitely are eligible for day-care centres.
Day-care centre, also called day nursery, nursery school, or crèche (French: “crib”), institution that provides supervision and care of infants and young children during the daytime, particularly so that their parents can hold jobs. Source
As you can see in the quoted text, they can be alternatively called as a nursery, nursery school or a child-care centre. But these usages would not mean that they are going there after school. Usually nursery school is attended before going to an actual school (more like a preschool you can say).
If they are visiting the place after school hours, they maybe called after-school care. These are applicable in the case of both primary schools and secondary schools.
Some programs are run by a primary or secondary school, while others are run by externally funded non-profit or commercial organizations. After-school youth programs can occur inside a school building or elsewhere in the community, for instance at a community center, church, library, or park.
Leisure centre on the other hand, is a building containing a swimming pool and other places where you can play sports. (Note: This is a very typical British usage, as can be known by the spelling of 'centre'). Hence it wouldn't seem fit in this context, unless you are actually taking them for swimming practices. Source of meaning.