A written constitution only really protects (or affects at all) the things it very specifically enumerates. And when I look at the judicial tools we have that do bind the government (the ECHR for instance) they seem on the whole to make a good difference. A UK constitution that enshrined certain rights (healthcare, free speech, and so on) would make me feel a lot more secure about what future governments could do. It might also provide a better example than the American constitution in the respects it is lacking.