It used to be much more common for parents to hand over many of the day to day childcare tasks to the nanny, even if they weren't working or especially busy. My impression is that, for the majority of modern families, those who have nannies use them to complement the parents, not replace them.
Also, many royals who were raised the old fashioned way still have or had good relationships with their parents. Queen Elizabeth II was close to her mother and father. She and Prince Philip seem to have decent relationships with their own children, (including Charles). Frederik of Denmark has gone on record in the past about how he spent limited time with his parents as a young child, instead being cared for by his nanny, and yet he's close to his parents today, as is Prince Joachim. It's become less fashionable these days to have your children "raised by nannies" but there's no evidence this method of bringing up children was ineffective, either.