"Even if someone managed to invent a robot nanny that kept kids safe as well as a human being, I wouldn’t want it to take care of my baby. I bet you wouldn’t either." This glosses over the purpose of child care -- it's more than safety, it's socialization, learning, etc. In a world where some level of automation (or offshoring) is possible for these services, I could definitely see some adoption. Think "Diamond Age."
It would be a status symbol to not have it, of course - real organically raised human children! - but the struggling middle class would pick up robonanny-as-a-service immediately if it was cheaper than other childcare.