No and No
As this article states
There are a number of associated characteristics of Britain’s unwritten constitution, a cardinal one being that in law Parliament is sovereign in the sense of being the supreme legislative body. Since there is no documentary constitution containing laws that are fundamental in status and superior to ordinary Acts of Parliament, the courts may only interpret parliamentary statutes. They may not overrule or declare them invalid for being contrary to the constitution and ‘unconstitutional’. So, too, there are no entrenched procedures (such as a special power of the House of Lords, or the requirement of a referendum) by which the unwritten constitution may be amended. The legislative process by which a constitutional law is repealed, amended or enacted, even one dealing with a matter of fundamental political importance, is similar in kind to any other Act of Parliament, however trivial its subject matter.
Basically since the English Civil War, there has never been any attempt or desire or even ability for parliament to introduce a formal constitution.
Most of the constitution is done by convention, again quoting from the article:
Another characteristic of the unwritten constitution is the special significance of political customs known as ‘conventions’, which oil the wheels of the relationship between the ancient institutions of state. These are unwritten rules of constitutional practice, vital to our politics, the workings of government, but not committed into law or any written form at all. The very existence of the office of Prime Minister, our head of government, is purely conventional. So is the rule upon which he or she is appointed, being whoever commands the confidence of the House of Commons (the majority party leader, or head of a coalition of parties).
Since as the parliament website says
no Parliament can pass laws that future Parliaments cannot change