names and titles or signing your name

So how does a British royal sign their name ...  well, they sign their first name  and that's it.

William does not sign William of Wales or William Wales.  He signs William.  His title is HRH Prince William of Wales because his father is the Prince of Wales.  The York princesses sign their names as Beatrice and Eugenie, except perhaps when handing in papers at school.  They might use York as a last name, but York will not be used a surname at the time of their weddings. 

The children of the Prince of Wales and the Duke of York have territorial designations as a part of their titles.  The two York princesses will lose the territorial designation when they marry.  HRH Princess Beatrice, Mrs so and so, for example.  The two Wales princes will also lose the Wales territorial designation as a part of their official style when Charles succeeds to the throne.

William will become HRH The Duke of Cornwall and Harry will be styled HRH The Prince Henry if he has not received a dukedom prior to his father succession.

William's children will be styled as Princes and Princesses of Cornwall until the time comes when Charles III creates his son as Prince of Wales .. and then ... William and Catherine's children will be styled as HRH Prince or HRH Princess christian name of Wales.

The surname Mountbatten-Windsor will be used on marriage registrations.  Anne, Andrew and Edward all had the surname Mountbatten-Windsor used on the registries.  Charles did not, which may mean William, as second in line, may not have the surname used on the marriage registration.  (We won't know this until April 29.)

The Mountbatten-Windsor surname will most certainly appear on Beatrice and Eugenie's marriage registrations.   

The situation is different for the children of TRH The Earl and Countess of Wessex.  Although Louise and James are legally entitled to HRH Princess and Prince, they have the rank of the daughter and eldest son of an earl.   This was announced at the time of Edward and Sophie's marriage.  Their children would use the surname Mountbatten-Windsor.  If the children were using royal titles, they would be HRH Princess Louise of Wessex and HRH Prince James of Wessex.

After Louise's birth, Buckingham Palace announced that she would be styled as The Lady Louise Windsor, but her surname would be Mountbatten-Windsor.  Her younger brother bears the courtesy title Viscount Severn, so when he is able to write his name, he will use James Severn, although his family name remains Mountbatten-Windsor.

There will be few changes for Edward's children when he receives the newly created Dukedom of Edinburgh.  He and his wife will become TRH The Duke and Duchess of Edinburgh.  Lady Louise Windsor's status will change from eldest daughter of an earl to eldest daughter of a duke.   James will take on a new courtesy title.   It is possible that the earldom of Wessex will become the secondary title, and Viscount Severn for the eldest son of the eldest son.  The new dukedom won't be created until Charles is king, and his father has died.

If Louise and James were accorded their royal status, they would move from "of Wessex" to "of Edinburgh."   They are styled not as royals but as children of an earl.  When Edward is created Duke of Edinburgh, his children will move to the style and rank of a duke's children.  (This will not affect their precedence at court.) 

Wales and York are territorial designations, and not official names.  Lady Louise Windsor is how she is styled by the court, although she will use Mountbatten-Windsor as a surname on official documents, but she will most likely sign as Louise Windsor.   James will not be a royal duke. He will be His Grace the Duke of Edinburgh and he will sign his name as Edinburgh as peers of the ream sign only with their title name.  His wife will be christian name Edinburgh.  This is how peers and their spouses sign their names, even on a check.   All the peers of the realm have family names (surnames).  James' children will have the surname of Mountbatten-Windsor.   I expect James' daughters to be styled as The Lady christian name Mountbatten-Windsor and younger sons as Lord christian name Mountbatten-Windsor. The eldest son will have a courtesy title.  

I think the Queen decided to allow Louise to be styled as The Lady Louise Windsor, and not The Lady Louise Mountbatten-Mountbatten was due to two factors: Louise is the granddaughter of a sovereign, and the standard use "of Windsor" is maintained as a part of the title.  The difference between Lady Louise and Lady Helen and Lady Gabriella is the surname.  Louise's surname is Mountbatten-Windsor, and not Windsor.

It also should be noted that the British sovereign is the fount of all honours.  She does not need to issue a letters patent or a decree regarding titles.  A press release will do.     Louise and James will never be styled as HRH Princess Louise and HRH Prince James.  They will grow up largely out of sight  - and out of mind - which is what their parents want.  They will have careers and will not be carrying out royal engagements.

It also will be interesting to see how William and Catherine's first child is styled if that first child is a girl.  (I do not expect Parliament to promulgate legislation to allow for the first born child, regardless of sex, minutes after Catherine gives birth to a daughter.)   According to the 1917 Letters Patent, only the first born son of the first born son of the Prince of Wales is entitled to the HRH and Prince or Princes.  The other children will be Lords and Ladies.  Thus, if Catherine gives birth to a daughter, Victoria, and then to a son, George, during Queen Elizabeth II's lifetime,  Victoria will be The Lady Victoria Windsor (or Mountbatten-Windsor) and George will be HRH Prince George of some territorial designation.   Victoria would move to HRH Princess Victoria as soon as Grandpapa Charles becomes king.  

 However, I won't be surprised if Queen Elizabeth II upgrades all of William and Catherine's children to royal so there is an equality of titles.