Re: That's not actually possible...
You do have rights, which come from a lot of different places - case law statute, the EU. But you don't have a single document - a constitution - a bill of rights - which lists all of your individual rights from start to finish.
This seems to say:
(1) a constitution is a bill of rights (in a single document)
(2) the UK doesn't have a bill of rights (in a single document)
(3) therefore the UK doesn't have a constitution.
The problem is that premise 1 is false. The equation of a constitution with a single document comprising a bill of rights is entirely your own invention. Although such a document might correctly be called 'a constitution' you haven't demonstrated that the only thing that can be called a constitution is such a document.