Children of Prince Harry and Meghan Markle will not be prince or princess but instead will be lord or lady – unless Queen change it. Great-great grandfather of Harry, king GeorgeV was the one who limited the titles inside the royal family, by issuing in 1917 a letter patent.
The letter reads: “…the grandchildren of the sons of any such Sovereign in the direct male line (save only the eldest living son of the eldest son of the Prince of Wales) shall have and enjoy in all occasions the style and title enjoyed by the children of Dukes of these Our Realms.”
In this case it means that the children of Harry and Megan if they will have any title until Elizabeth II is on the throne its not going to be prince or princesses, but instead they will be called as Lord or Lady Mountbatten-Windsor. Queen could write a new Letter Patent to change this, as she did for the children of Duke and Duchess of Cambridge.
From the other hand George was entitled as Prince George of Cambridge as he was the oldest son of the oldest son of Princes of Wales. The same happened with Princess Charlotte of Cambridge.
When Kate was four months pregnant Queen Elisabeth issues a Letter Patent under the Great Seal of Realm saying that “all the children of the eldest son of the Prince of Wales should have and enjoy the style, title and attribute of royal highness with the titular dignity of Prince or Princess prefixed to their Christian names or with such other titles of honour”