Both the US and UK operate under common-law systems with substantial judicial precedent. In practice, there is very little difference between the two systems.
I'm not a lawyer, but I always associated (perhaps wrongly) common law with common sense. I don't see much of the latter here.
Otherwise the prosecutor can stand up in court and say you were "uncooperative" and resisted, even obstructed attempts to investigate the crime, and if your lawyer tries to frame this as you exercising your right, the judge will not permit this because you didn't utter the magical incantations.