The UK does have a constitution, it's just not a single codified constitutional document. This has led to some really interesting constitutional documents, for example, a letter to The TImes under a pseudonoym Senex was called a constitutional document in evidence taken before the Justice Committee:
> Q16 Chairman: Is the letter to The Times in 1950
under a pseudonym Senex, which we now know is
Alan Lascelles, a constitutional document that now
guides us?
> Lord Turnbull:In a strange way,it is,yes; people have
accepted the logic of the arguments that he put
forward.
Your last note shows what the UK lacks - constitution as a supreme document.
In UK, the Parliament is the supreme authority, and can amend that unwritten constitution simply by passing an Act of Parliament, which only requires a simple majority.
In effect, this means that the Parliament is not really bound by the constitution. Which makes it largely useless for the purposes for which it is heavily used and relied upon in US.
> Q16 Chairman: Is the letter to The Times in 1950 under a pseudonym Senex, which we now know is Alan Lascelles, a constitutional document that now guides us?
> Lord Turnbull:In a strange way,it is,yes; people have accepted the logic of the arguments that he put forward.
From https://www.publications.parliament.uk/pa/cm200910/cmselect/...
(Incidentally that letter has been superseded by the Fixed Parliament Act).