In northern Cameroon, babies generally wore a waist-string and a set of protective amulets and not much else. Babies spend most of their time in a sling on the mother’s back, so the mother becomes very acquainted with the child’s patterns and signs and would generally hold the baby over a bowl (with the waste disposed of in the latrine). The rest of the time, the baby roams fairly free-range in the living space, but since the living space was usually a shaded yard with a gravel or (for the very poor) dirt floor, it wasn’t too hard to clean up any messes. That said, I’ve been peed upon by more than one baby!
In much of China, the situation is similar. Parents become acquainted with the child’s patterns, and the child learns to respond to a low whistle. As the children get into their toddler years, they wear the characteristic “split pants” that allow them to squat and do their business wherever. It’s not unusual to see a child being encouraged to defecate into a plastic bag in a public spot.