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The exact details of Britannia's growth up until the modern period is only heard in bits and pieces in the entire Code Geass universe, and not entirely in the anime.
During the reign of Elizabeth I (which would presumably be dated as 1612-1657 a.t.b rather than 1558-1603 AD), the queen remained unmarried, but bore a son, the father of whom is never publicly identified, though candidates were Sir Robert Dudly, the 1st Earl of Leicester, Sir Robert Devereux, the 2nd Earl of Essex, and Sir Carl, the Duke of Britannia. Each of these men would use this knowledge to garner influence and power in the country. Elizabeth I's son ascended to the throne as Henry IX upon the death of his mother in 1657 (our 1603). His ascension was said to mark the beginning of "the Golden Age of the Tudor dynasty." (In actual history, Elizabeth I died childless as the last monarch of the Tudor dynasty.)
Britannia's history drastically diverges from our timeline during the "Age of Revolutions," starting with the failure of the American Revolutionary War in the 1830s (our 1770s), which is known as "Washington's Rebellion" in the history of Code Geass. The revolution failed because Benjamin Franklin was bribed off by the Duke of Britannia with the title of earl. George Washington and the Continental Army suffered a crippling defeat at the battle of Yorktown, at which point control of the North American colonies by Britannia was assured.
The colonial possessions of "Britannia" at this time are not clear in relation to our British Empire; it is not made clear if Britannia ever colonized the Indian Subcontinent (as China controls it at the beginning of the series) or if it was ever an active colonial power in Africa. It seems that Britannia's most important overseas possessions were in North America.
