According to Buckingham Palace, the announcement that Prince Edward's children would not be Princess and Prince was legally binding.
That is what I was told when I wrote and asked BP. All that is required for a person to gain or lose a title within the royal family is for The Queen's Will to be made known and the Queen has done that with regard to Louise and James.
[...]
The letter said that Louise and James are not HRHs as The Queen's Will, that they aren't holders of that styling, has been made known via the announcement made on Edward's wedding day.
The letter is as follows:
Dear xxxxx (sorry not making public my name)
Thank you for your request for clarification about the question of the styling of the children of HRH The Earl of Wessex.
You are correct in your interpretation of the announcement made in 1999.
The Queen's Will was made known on HRH The Earl of Wessex's wedding day and as such none of his children do now, nor will in the future, have the style of HRH Prince or Princess. As Her Majesty is the fount of all honours all that is needed for a style to be given or taken, except for a substantive peerage, is that Her Majesty's Will is made known.
Thank you for your interest in this subject.
[...]
The announcement also stated the agreement to create Prince Edward Duke of Edinburgh after the deceases of both of his parents.
Title of HRH The Prince Edward
The Queen has today been pleased to confer an Earldom on The Prince Edward. His titles will be Earl of Wessex and Viscount Severn. The Prince Edward thus becomes His Royal Highness The Earl of Wessex and Miss Sophie Rhys-Jones on marriage will become Her Royal Highness The Countess of Wessex.
The Queen, The Duke of Edinburgh and The Prince of Wales have also agreed that The Prince Edward should be given the Dukedom of Edinburgh in due course, when the present title now held by Prince Philip eventually reverts to the Crown.
The Queen has also decided, with the agreement of The Prince Edward and Miss Rhys-Jones, that any children they might have should not be given the style His or Her Royal Highness, but would have courtesy titles as sons or daughters of an Earl.
So Edward's children are known as James, Viscount Severn and Lady Louise Windsor,
Prince Edward's son is formally known as simply Viscount Severn, without "James".
https://www.royal.uk/succession
Apart from divorcées (e.g. Sarah, Duchess of York) and widows, a person who uses a title from the British peerage is not formally addressed with their first name.
Edward was not offered DoC as far as we know.
Although there was no official confirmation, there were reports in the press to that effect.
Royal wedding: Prince William asks the Queen not to make him a duke - Telegraph
"Prince Edward was going to be the Duke of Cambridge, but he watched the film Shakespeare in Love, which had a character called the Earl of Wessex," says the courtier. "He liked the sound of it and asked the Queen if he could have that instead."
James and Louise could already be HRH Prince James of Wessex and HRH Princess Louise of Wessex like his two elder brothers' children. However they are styled as children of an Earl, as was the wish of the parents. See the wish of the Duke and Duke of Sussex that their son Lord Archibald Mountbatten-Windsor, Earl of Dumbarton, is not known as such. A personal choice can overturn an official style.
Yes, but only with the Queen's agreement. Princess Eugenie has chosen to continue styling herself Princess Eugenie of York to this day, as shown on her professional and charity websites, but several months ago the Court Circular began referring to her as Princess Eugenie, Mrs. Jack Brooksbank.
Archie still has his titles, he is legally Earl Dumbarton and once Charles is King he will be an HRH. They just said at this time he is not using his titles.
The legal Earl of Dumbarton is the Duke of Sussex (Prince Harry), but his son, who is legally untitled, will have the choice of using this title by courtesy in accordance with tradition.