I don't know about grade schools because my kids went to a near north side magnet school, but for pre-K I sent mine to the Children's Developmental Institute, at 71st and Stony, which I think has a branch in or near the Beverly area. I know the name sounds institutional, but the staff there were loving and caring and the kids have a lot of fun. It's a very active program. The academics don't matter much at that age; they won't hurt but mostly what the kids need is socialization and things to do, which they will get.
Don't discount the possibility of a public school plus a Park District after-school program for kids K and older. You can check first, but many schools will pick up the kids at home and then bus them to an after-school program. That's what I did when my family became to numerous to afford private schools, and I was very happy with the arrangement.