Why didn't Prince Henry, Duke of Gloucester become king when George VI died? By everything that I know, as the next male in succession, George V's third son should have become king after the first abdicated and the second died.
The type of succession that you are referring to is called agnatic primogeniture. Inheritance according to the seniority of birth amongst the of sons of a monarch.
King George V had five sons. Prince Edward (future King Edward VIII), Prince Albert (future King George VI), Prince Henry, Duke of Gloucester, Prince George, Duke of Kent and Prince John, who died as a child. After King George V's death, in 1936 and King Edward VIII's abdication in 1937, the next son came to the throne as King George VI. In 1937, at the start of King George VI's reign, the first four places in the line of succession were occupied by Princess Elizabeth, Princess Margaret, Prince Henry, Duke of Gloucester and Prince George, Duke of Kent.
The succession to the British throne is by male preference primogeniture. Males have precedence over females, however if there is a female she is not excluded. Because of this, King George VI's elder daughter, Princess Elizabeth became the sovereign and not Prince Henry. If the King died without children, then Prince Henry would have become king in 1952 because he was next in line. If this had occurred then after King Henry IX's death in 1974, his son, Prince Richard, Duke of Gloucester would be king, possibly reigning as King Richard IV.
It is interesting to note that had King Edward VIII not abdicated and not had children, the succession would likely look as it does today. The only difference is that the present Queen would have come to the throne upon the death of her uncle in 1972 instead of 1952 when her father died.
© Marilyn Braun 2011