Is there a known style guide or documented set of stylistic rules for the English language that Queen Elizabeth II and/or her office uses in emitted correspondence? Is there any way to know how QE2 would write something without digging through old correspondence for similar uses?
Queen Elizabeth II's government — The Prime Minister and the cabinet — write her official speeches. As explained here: Queen's speech
Nor is she the author of her official correspondence, according to the official British Monarchy website. (Source)
The office of the Private Secretary, supports the Queen in her job as Head of State. The current Private Secretary to The Queen is The Rt. Hon. Sir Christopher Geidt KCVO OBE who was appointed to his role in September 2007.
The Private Secretary is the channel of communication between the Head of State and the Government, not only in the United Kingdom but also in the 15 other realms of which The Queen is Sovereign.
Other responsibilities include organising The Queen’s official programme at home and overseas; liaising with the Households of other members of the Royal Family; and dealing with The Queen’s official correspondence and correspondence with members of the public.
The Private Secretary prepares The Queen’s speeches and messages, and his/her office arranges photographs and official presents, portraits and messages of congratulation.
An exception to the rule is the Queen's Christmas speech which the BBC claims:
The Queen writes her own Christmas speeches and it is one of the rare occasions when she does not seek government advice and is able to voice her own views.
The Guardian confirms that the 2012 Christmas speech was by her own hand.
This year, her message, which she writes herself, will highlight the "splendid summer of sport", and pay tribute to the nation's Olympic and Paralympic athletes, who gave spectators the chance to revel in the "excitement and drama".
Therefore, in order to know which style guide (if any) is employed, one would need to ask David Cameron, all the ministers of the British cabinet and finally, the Queen's private secretary Sir Christopher Geidt themselves.
If it is of any help, the 47-year-old Prime Minister, David Cameron, attended Eton College before studying Philosophy, Politics, and Economics at Oxford university, so presumably he is well-versed in the Queen's English.