Do we need a Fourth Amendment?

Yesterday, I read a suggestion that there ought to be legal protection in the UK which is equivalent to the Fouth Amendment of the US Constitution. The Fourth Amendment states that:

"The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized."

The argument put forward was that with Fourth Amendment style protection, the Natwest Three would not have been able to be extradited as it would have required probable cause to be established in a UK court, which, if my understanding of the case is correct, would not be possible as from our perspective no crime has actually been committed.

Could this be the kind of thing that Cameron was thinking of for our own Bill of Rights (or an amended Human Rights Act)? I've just been looking through the current Human Rights Act and the European Convention and I can't see any equivalent protection to the Fourth Amendment (willing to be shown otherwise of course).