Harry’s children will be Prince and Princess won’t they? Andrew’s were.
Not at the moment, unless the Queen allows a change as she has for Charlotte and the new baby.
Only the grandchildren of the monarch on the male line are entitled to the HRH Prince/Princess, plus the eldest son of the eldest son of the Prince of Wales.
So, until the Queen changed it George would have been HRH Prince George, but Charlotte and the new baby would have been Lady Charlotte and Lord Whatever. Then when Charles becomes King they'd become Grandchildren of the monarch on the male line so would be HRH Prince/Princess.
This is where Harry's children are currently at. They'd be Lord/Lady Windsor until Charles becomes King unless the Queen issues LPs.
same as Anne's (Zara and Peter), they were entitled to the title Prince/Princess, but their parents decided not to give them the titles.
Anne's children weren't entitled to Prince/Princess automatically as only male line grandchildren are. Hence why the Queen offered Mark Philips a title.
Back when the Queen had her children her father had to issue letters patent to allow them to be HRH Prince/Princess otherwise they'd have been Lord/Lady Windsor.
Edward's children are an anomaly because legally they are entitled to be HRH Princess Louise and HRH Prince James, but the statement from the Queen said they would be styled as children of an Earl. Which is basically the same as Camilla, who is HRH The Princess of Waless, but is styled as HRH The Duchess of Cornwall.