Actually she wont be Princes anything of anywhere, she isnt a princess and neither was Diana. She will be titled, like Diana was, as Name she choses to use (be that Meghan or Rachael) , Princess of Wales.
Two entirely different things Princess X of Wales, and X, Princess of Wales. Best illustrated by the Yorks, who are princessess. Princess Beatrice of York. She is a princess in her own right, it's not an acquired title by virue of marriage, like Diana.
If you want to really formal, Harry is: The Prince, Henry, of Wales as opposed to The Prince of Wales, Charles
Royal Dukedomes are even better
. Prince Michael of Kent is a second son, therefore only a prince, whilst his elder brother Prince Edward, but known as the Duke of Kent, but no-one ever calls him Prince Edward, similarly no one ever calls the Duke of Gloucester, Prince Richard