Interesting question. Here's my two pennies.

I think you might be confusing the reasons for the English Civil War (or the Wars of the Three Kingdoms) with the reasons for conflicts such as the French Revolution, which spawned constitutions such as the Declaration Of The Rights Of Man.

The French Revolution saw the birth of the nation-state and a sovereign people. The underlying reasons are obviously hotly debated, but in essence the ancien regime was overthrown because vast swathes of the French elite lived outside of the law, oppressed the peasantry etc. it was very much a national, populist uprising rather than a parliamentary one.

The Wars of the Three Kingdoms began primarily due to the actions of a select few people - namely Charles I and John Laud. It was parliament's disagreements with Charles that brought about Civil War, exacerbated by his reckless financial policies and confessional differences, and had little to do with widespread rejection of the English way of life.

Ergo, the Wars of the Three Kingdoms did not represent a sea-change in national identity that necessitated a completely new method of government with a different set of laws as with France/America/Italy, merely a rejection of one man's rule that manifested in an anti-monarchical sentiment. In reality, that sentiment didn't last very long and pretty soon after Cromwell's death England had restored the monarchy.