So here's an article on the status and title of Meghan and Harry's impending baby.
I know. You don't care. But in a world full of pressing cares, it's the fact that this is of no significance whatsoever that makes it refreshing to talk about.
According to the article, the baby will not automatically be designated a prince or princess. I think that's right. Among the Queen's cousins, the title of prince or princess goes down only two generations from the monarch. Whether the blessing will automatically descend upon them if and when Prince Charles becomes king, I'm not sure but I think so.
However, the article also says that Kate & Wills's children had to be individually given that status: they didn't get it automatically. That may be true for Charlotte and Louis, but the order of George V limiting the use of prince/princess is quoted in the article as not applying to "the eldest living son of the eldest son of the Prince of Wales." That describes Prince George. He gets it automatically, so allow me to point out that on this point Wikipedia is right and the Washington Post is wrong.
However, that's not the wrongest. This is about the title of the children of Meghan and Harry. The article says "It is believed that any children of the duke and duchess of Sussex will be known as Lord or Lady Mountbatten-Windsor." Believed by whom? Only by people who don't know the nomenclature of British nobility.
The eldest son (as the patent of Harry's duchy is the usual males-only) will be known formally as the Earl of Dumbarton, by the customary rules that the son and heir of a senior peer takes his father's highest subsidiary title by courtesy.
Other children will be known as Lord or Lady First-name followed by last name, not with last name immediately following title. See Lord Randolph Churchill in history and Lord Peter Wimsey in fiction. There's no such thing as Lord Last-name in British nomenclature, only Lord Title, and "Mountbatten-Windsor" is nobody's title. (I believe you can be Lady Last-name, but only as wife of a knight, not as part of the peerage.)
It is true that the children's legal surname will be Mountbatten-Windsor, but it's very common for people with double-barrelled last names to employ only one barrel of it in their use-names. Winston Churchill's actual surname was Spencer-Churchill (yes, he was a distant relative of Princess Di), but neither he nor his father (see above) nor any of his descendants have been known that way. The one Mountbatten-Windsor in the Lady First-name position is Prince Edward's daughter, who is styled Lady Louise Windsor. Quite possibly Harry's children will be styled likewise.
And that's the straight dope as far as I know it.