Played onscreen by magnetic newcomer Madina Nalwanga, the Queen of Katwe's story begins in 2007. Her father has just died and her mother, Harriet (Lupita Nyong'o), can barely keep the family fed.

Young people ponder their moves at the Som Chess Academy in Katwe, Uganda, on a recent afternoon. Phiona Mutesi, the prodigy whose story is told in Disney’s “Queen of Katwe,” started playing ...