It'd be great if the UK had a written constitution. But let's also be realistic here. The US also has a written constitution and the NSA has a database exactly like this one anyway, and such databases are also available to a mishmash of random law enforcement bodies. The constitution didn't help.

Also, the UK is signed up to the ECHR which theoretically guarantees a right to privacy. It's sort of like the US Bill of Rights except useless, because it was drafted by Europeans so every right has a giant get-out clause. In this case the so-called "right" to privacy exists only as long as it doesn't conflict with the "needs of a democratic society". That sort of thing crops up all the time in this document.

It's not sufficient to have a constitution. It must have teeth as well.