Well I got started too late in the day to present a fleshed out version. So let this serve as an abstract.
Britain has never had a written constitution, instead it had what latter lawyers called the 'Ancient Constitution' which was both older than memory or even history reached and was in form perfect. Being perfect it was not changed or indeed changeable. Moreover it was not in origin considered to be the King's Law, it was instead before and to some degree above the King.
Now Britain did have royal formulations of law which were drawn up in what are now known as Law Codes, starting with that of King Aethelred of Kent of King Ine of Wessex which latter were incorporated in the latter Code of Alfred the Great. Alfred's Code is not designed to be foundational, instead he has gathered what he liked of other codes and gathered them together with some modifications. The best way to read Alfred's Code is as a collection of Administrative and Legal Regulations particularly as governing the King's rights to exercise jurisdiction. Which set off a struggle that would extend for centuries and indeed were not settled until the late 19th century, that is who was in control of the judgement in legal cases. Eventually it all ended in the hands of the King's Judges but this was a long process.
But none of this fundamentally was seen to change the Ancient Constitution, to the end English people held that they had rights against the King and particularly in the area of property and just as importantly in taxation. Certain taxes (or so we would call them) were due to the King as right, and others to the King as landlord, but at no time in English history was the principle that the King could just set taxes at will. And attempts by various English Kings to claim expanded rights were resisted and often enough by force.
The Magna Carta, often thought of as the basis for latter Anglo-American jurisprudence, is the product of just such a rejected claim. King John made demands that challenged the property rights of Lords and free-holders which led to a Civil War. Which John lost. The result was that the opposition laid forth their case, based on the Ancient Constitution (though at that period they didn't call it such), that John must renounce certain specific claims. In 1215 John was forced to submit. This was not a one time event, the Great Charter was reissued many times and serves as a cornerstone of the British Statute Book, but does not serve as the foundation for the constitutional system.
My suggestion is that the Modern American Conservative movement draws from this now eight century struggle to contain the King and the Government into their proper role and that insists that certain rights existed prior to any Kings, or at least any earthly Kings. To that extent the line between the Ancient Constitution and Natural Law begins to blur thus explaining why Conservatives seem to have the selective acceptance of the American Constitution, to the degree that it violates previous rights and laws it is only partially binding.
I don't know that I am totally persuaded by my own argument here, and I will try to continue developing it. For what it is worth consider this version 1.0