Our constitution over here comprises the entire body of Statute and Common Law, and has been constantly evolving since Aethelstan was delcared King of the English in 927 CE.
Every case decided in the courts either depends upon or sets precedent, and these precedents compose the Common Law. Statute Law is composed of all currently active Acts of Parliament, as amended.
Thus our Constitution evolves to accommodate change, albeit with some delay at times!
The only basic rule is that all authority resides in the Crown (the institution, not the person of the monarch). The Judiciary, the Armed Forces and the Civil Service are all servants of the Crown,not the government. Parliament draws its authority from the Crown, but gains its mandate from the people.
Please don't misunderstand Brexit. The aim was, and is, to secure a return to the terms under which the UK entered the Common Market, as a partner in a free trade area. What the people of Britain rejected was the proposed political union, the Federal Europe agenda.