How does Great Britain end up with a Codified Constitution in the present day?.

The most likely point is after the Glorious Revolution. A lot got officially laid down at the time, and it's not hard to imagine a legal equivalent of the Domesday book put together so that the foreign king and his Parliament know where each other stand.
 
Easy: Have a royal screw up in the early 19th century with his mixture of incompetence and reactionary views leading to a British Revolution and a republic being proclaimed and the first thing said United Republic of (Great) Britain does is write a written constitution.
 
Easy: Have a royal screw up in the early 19th century with his mixture of incompetence and reactionary views leading to a British Revolution and a republic being proclaimed and the first thing said United Republic of (Great) Britain does is write a written constitution.

You don't need take monarchy away. Many monarchies have constitution. One easy way might be that it is written soon after Union Act.
 
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