As it stand unless Queen Elizabeth changes letters patent again, which I think she would have done already, when Charles becomes King they legally become Prince and Princess like Louise and James Wessex. They are styled as children of an Earl because their parents knew they would never become working royals. I imagine Charles thought Harry and whoever his wife would be would be sensible like this and style any children as children of a Duke. Meghan will obviously not do this. Even Sophie has said when her kids are 18 they can choose themselves to be styled as a Princess and a Prince but she can't see it happening.Are titles given in an official royal ceremony like when they knight people? Or does the Queen just sign a paper and that is it?
I actually agree with a law that would require any prince/Princess to live and be educated in the UK. How can you expect to be Prince of somewhere but don’t even know the history, language and culture of the place? Harry’s kids won’t know Buckingham Palace from Westminster.
There is sometimes a ceremony when the Prince of Wales title is bestowed upon the Duke of Cornwall of the time as that is not automatic and granted by the monarch. Edward VIII and Charles both had these in Wales