KAMPALA, Uganda—On a potholed road next to an open sewer, four men faced off in a Kung Fu battle. A khaki-clad soldier ran up, firing a machine gun fashioned from old lawn-mower parts. Two cannibals caked in mud and dried grass approached the bloody corpses and licked their lips.

Welcome to Wakaliwood, a movie studio and production house that is riding a wave of popularity in East Africa and drawing a cult following overseas, with crowded screenings in Hong Kong, Stockholm and New York. Based in a ramshackle Kampala slum called...