I will argue that the UK doesn't have a Constitution and that's because of this one thing: No parliament can bind a future parliament. A Constitution is exactly this: a contract binding all future Parliaments to a specific set of axioms that must be respected.
The US constitution is very similar, except in two important regards: amendments require two thirds majority votes in both houses and ratification by 75% of the states.
We don't have the state mechanism. You could argue the four nations could serve a similar purpose, though there's a debate about how democratic that is when England makes up something like 85% of the UK population (and doesn't have its own legislature).
