Yeah we do.
Multiple, technically.
The 1689 Bill of Rights even inspired the American Bill of Rights.
The main issue is we have a legislature (House of Commons) with supremacy over every other body of government. It'd be like if the House of Representatives had supreme say over the entire United States instead of the rigamarole of the Senate, the President, the Supreme Court, and the House all operating in conjunction with one another. If that was the case, whoever had a majority in the House could just undo parts of the constitution, which'd probably fuck everything up?
Not to defend my country, but arrests are defined in America is actually taking people in custody, right? In the UK an "arrest" also includes merely "investigating" the matter. If you're issued a warning, a fine, told to identify yourself, told to attend the police station of your own free will, etcetera, that counts as being "arrested" in the UK. If you're told to pull over by the police and get issued a warning for speeding or unsteady driving, you've been "arrested". Yes, it's retarded. No, it doesn't mean hundreds are being thrown into prison everyday. Yes, the fact people are being warned at all is an injustice. The UK police are fond of this because it enables them to get their numbers up which justifies funding so they'll seize any chance they can get to issue a fine or pay someone a visit, it's free money.