This is one of the advantages of having a written constitution. You can't just do things on a whim that violate basic principles of the country, since they're clearly codified in writing.
As for doing things on a whim that violate basic principles of the country, when it comes to constitutionally-mandated protections against unreasonable search and seizure, I'd rather go through UK airport security than US. Then there's oddities like 'free speech zones', where you can say what you like, but only where you're told to. The second amendment is hardly 'clearly codified' - it's highly ambiguous, yet has such significant impact on US society.
The US consitution has some great stuff in it and was a watershed document, but it's not a magical shield simply because it's written down on a document named 'constitution'.
Moreover, the UK would then get a lot of shit overseas, particularly in America, for being an example of how European-style socialism leads to a lack of civil liberties, and that might put pressure on the British to actually codify that right.