Prince William, duke of Gloucester is another example of the son of a princess. He was born in 1689 during the reign of his aunt Mary II and her husband/first cousin William III.
The sons of princess Margriet of the Netherlands have the style of His Highness and the title prince of Orange-Nassau.
The children of HRH princess Madeleine of Sweden were born HRH prince(ss) of Sweden but later like the children of their uncle Carl-Philip were demoted in an effort to slim down the Royal Family and give them the opportunity to live a more normal life.
In Spain the children of both sisters of king Alfonso XIII were Infante/Infanta.
Sebastian Gabriel de Borbon y Bragança, son of luso-spanish Infante Pedro Carlos of Borbon and Infanta D.Maria Teresa of Bragança, Princess of Beira, eldest daughter of King D.João VI, was granted the title of Infante of Spain AND Portugal with the style of HH.
Infante D.Pedro Carlos lived most of his short life in Portugal and accompanied the PRF to Brazil when the french napoleonic troops invaded the country, where he died.
Infante D.Sebastião was born in Brazil.
I know that the Fife sisters (Alexandra and Maud) were accorded the title Princess, with the qualification of Highness, by the 1905 Letters Patent of their maternal grandfather, King Edward VII of Great Britain, in the middle of his reign. Similarly later on, in the 1948 Letters Patent, King George VI declared that all children born to his elder daughter, the then-Princess Elizabeth, would be accorded the title of Prince(ss) with the qualification of Royal Highness -- even if born within his reign.
Otherwise, the Fife sisters would have continued to be styled with the appellation Lady, which is what the Princess Royal would have been, as a daughter of a duke. And the Prince of Wales would have been the Earl of Merioneth.
Are these the only instances in the royal history of England or Scotland, whereby cognatic grandchildren of a reigning monarch were given the princely title?
I know that this was known to happen all the time in Spain -- as long as a princess remained in her own country and married with approval.
The sovereign is the fount of all honors, and it is certainly his prerogative to grant royal titles -- as one saw in 1995, when King Albert II of the Belgians (the then-reigning monarch) declared his son-in-law a prince of Belgium.
What about Portugal? Have cognatic grandchildren of reigning monarchs been given royal titles there, too?