In a parallel world where super-capable nannies where mass-produced, capable of taking care of kids and even educating them, how would this affect how families work in this world?
If the nannys are the primary caregivers from birth, and they are built like Cosworth, you can expect big problems. Babies need to cuddle. Metal doesn't cuddle.
The classic research along these lines goes back to Harlow in the 60's, and a starting point to read about it is here
Infant rhesus monkeys were taken away from their mothers and raised in a laboratory setting, with some infants placed in separate cages away from peers. In social isolation, the monkeys showed disturbed behavior, staring blankly, circling their cages, and engaging in self-mutilation. When the isolated infants were re-introduced to the group, they were unsure of how to interact — many stayed separate from the group, and some even died after refusing to eat.
So a surrogate mother (Cosworth) was provided
In this study, Harlow took infant monkeys from their biological mothers and gave them two inanimate surrogate mothers: one was a simple construction of wire and wood, and the second was covered in foam rubber and soft terry cloth. The infants were assigned to one of two conditions. In the first, the wire mother had a milk bottle and the cloth mother did not; in the second, the cloth mother had the food while the wire mother had none.
In both conditions, Harlow found that the infant monkeys spent significantly more time with the terry cloth mother than they did with the wire mother. When only the wire mother had food, the babies came to the wire mother to feed and immediately returned to cling to the cloth surrogate.
Harlow’s work showed that infants also turned to inanimate surrogate mothers for comfort when they were faced with new and scary situations. When placed in a novel environment with a surrogate mother, infant monkeys would explore the area, run back to the surrogate mother when startled, and then venture out to explore again. Without a surrogate mother, the infants were paralyzed with fear, huddled in a ball sucking their thumbs. If an alarming noise-making toy was placed in the cage, an infant with a surrogate mother present would explore and attack the toy; without a surrogate mother, the infant would cower in fear.