A beginning doomed to failure
The success of the U.S. Constitution is due, mainly, to the vision of the founding fathers who, on reflection, did a pretty good job, but it's significant benefit there was no political baggage surrounding the writers of the document.
Fast forward to 1982 when Canada's home written Constitution was signed into law by the Queen giving Canada full independence from the UK - British North America Act in 1867 governed Canada until this point.
Although the Canadian Prime Minister of the day, Pierre Trudeau, wasn't the most popular guy to hold that office he did have the foresight to get the constitution from a wet dream to reality. Examination of the Canadian Constitution will show there are quite a few 'outs' for political office holders that aren't to be found in the U.S. Constitution.
Jump to today's feeble attempt of a British 'Constitution' and it is completely devoid of protection of citizens that can be found in both the aforementioned constitutions.
If the UK government intended to protect the citizenry it would strike down many of the give-aways that Plod enjoys such as no-warrant searches, your-password-or four-years, entitlement not to answer police questions without penalty.
As both a citizen of Canada and the U.S. (and, actually, the UK) I enjoy many rights that the politicians would never dare give the British. I don't have to respond to Plod yelling: 'Oi, you' if I haven't committed a crime, I don't have to identify myself as a pedestrian (vehicle use requires ID) and I am free to take pictures of any damn thing I see.
The American Constitution can be amended but only with extreme difficulty; the Canadian likewise but with far lower requirements and the British attempt ...
Therefore, IMO, no UK 'constitution' will ever succeed unless politicians, with characters that we see no longer, drafted the document.