After thinking a lot about the score of this question, suddenly I think I have an enough strong idea to summarize it as an answer. Actually, the downvotes and comments to this question made for me a much more clear answer as the posts.
The essence of the other answers are obviously not acceptable; a sample of laws in the role of a constitution is clearly not a constition. That is only a sample of laws.
The reactions have clearly shown as I have touched - unintentionally - a sensitive point. I think some patriotism is okay to a country, but that is about the country, and not about its legal system.
There must be something, which is
- specific only to the UK
- might have some emotional importance is many politics SE users
- to the extent that they politely and "factually" try to "correct" me into a clearly faulty explanation.
The only what can be specific in the UK, is that they have 800 years old laws still in force (at least on paper) and no one has the spine to clean them up. No one wants to polish them up, they are pretty happy with it. The long-term stability of their legal system, consequently their political system, and is an admirably rightful source of their national pride.
My current best bet is that tradition is actually not so strong as they imagine, but all the players believe that the others would act upon it, and that makes also them to keep the tratition. It is some like a Nash equilibrium for them. And actually that is their real constitution and not a written paper. This is also the reason, why they do not write a paper: it would be a competitor to their tradition.
Thus, one thing is that they do not have a written constitution, I believe actually they can not have it - it would be a change what no one wants.
Instead, their "sample" of laws in the role of constitutuion, and the many century long political tradition fulfills the role of the constitution. As long it is fine to them, nothing will change that, and - as the votes and comments show - it is pretty fine for them.