Letters Patent, by which the sovereign of the United Kingdom regulates royal titles and exercises the 'Royal Prerogative', are the 'legal' way of doing things. This is why there was uncertainty resulting from the Press Release made at the time of the Wedding of TRH The Earl and Countess of Wessex and their future children's titles which stated an 'agreement' that:
any children of their marriage should not be given the style of His or Her Royal Highness, but would have courtesy titles as sons or daughters of an Earl.
But Louise and James are the Grandchildren of a sovereign. Any children of Meghan and Harry will, at the present time, be Great-Grandchildren of a sovereign and, as such, are governed by the Letters Patent of 1917 which regulate royal titles in general and restrict them to no further than the generation of grandchildren of a sovereign.
...hereby declare our Royal Will and Pleasure that the children of any Sovereign of these Realms and the children of the sons of any such Sovereign and the eldest living son of the eldest son of the Prince of Wales shall have and at all times hold and enjoy the style title or attribute of Royal Highness with their titular dignity of Prince or Princess prefixed to their respective Christian names or with their other titles of honour And We do further declare our Royal Will and Pleasure that save as aforesaid the style title or attribute of Royal Highness Highness ... and the titular dignity of Prince or Princess ... it being Our Royal Will and Pleasure that 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...
However, there have been two exceptions made, both because the monarch has lived to a 'ripe old age'.
Firstly in 1898 when the then Duke of York, who would become HM King George V, married and started to have a family which would be Great-Grandchildren of a sovereign, namely HM Queen Victoria. She extended royal titles to "... the children of the eldest son of any Prince of Wales..." In other words, all the children of the eldest son of any Prince of Wales, not just the sovereign's eldest or first-born son but any surviving eldest son who would be Prince of Wales. Unfortunately, although his own children enjoyed this privilege, King George V basically reversed this when he issued his Letters Patent of 1917!
The other exception was in 2012 when the present Duke of Cambridge started to raise a family and was in the same position as George V in 1898 but with the added aspect of the imminent implementation of the Succession to the Crown Act. (This Act came into force the following year.) As the regulations stood, Prince George would still have been Prince George, but a girl would have been Lady N Mountbatten-Windsor. With the Succession to the Crown Act doing away with male primogeniture, whereby a younger son takes precedence over an elder daughter, we could have had a Lady N Mountbatten-Windsor having precedence over a Prince, which, though legal, would have been rather absurd.
The Queen therefore issued Letters Patent dated 31 December 2012 to declare 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.
Although one would have thought something would have been done by now, there is an argument for doing something similar to cover Harry and Meghan's children and make them equal to William and Catherine's children, or not as the case may be if Harry and Meghan don't want their children to be royal. Yet...
The thing that the press seems to have forgotten is that, when Her Majesty passes away and Prince Charles comes to the Throne, Harry will cease to be HRH Prince Henry of Wales, Duke of Sussex and will become HRH The Prince Henry, Duke of Sussex. His and Meghan's children will cease to be Great-Grandchildren of a sovereign, but Grandchildren of a sovereign and will therefore, under the Letters Patent of 1917, be royal with the title Prince or Princess N of Sussex.
Any knock-on effect of William and Harry's children's present rank on heraldry will be minimal as the first child to be expected to be granted Arms is Prince George who will not be 18 until 2031.
Maybe The Queen will be her pragmatic self and leave everything to take its course as it is. Well will see...