I think you're approaching this from the wrong direction.

In essence, the rebellious colonists believed that they were entitled to the the traditional rights of Englishmen, despite the fact that they were no longer in England. (Blackstone's Commentaries on the Laws of England is the traditional text that sums up the state of late-18th century English law.)

Let's look at the Declaration of Independence for examples. Once you get past the famous preamble, the colonists provide a laundry list of legal rights that the King has refused to respect.

So,

He has dissolved Representative Houses repeatedly, for opposing with manly firmness his invasions on the rights of the people.

He has made Judges dependent on his Will alone, for the tenure of their offices, and the amount and payment of their salaries.

He has kept among us, in times of peace, Standing Armies without the Consent of our legislatures.

He has affected to render the Military independent of and superior to the Civil power.

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These are all specific legal rights that the colonists believed they had! And you'll notice, in reading the Declaration of Independence, that the indictment is not of the traditional laws themselves -- the Declaration, instead, is accusing the King of denying the colonists the rights they're entitled to under the law.

Eventually, after the colonials won the war, the worst of these abuses were specifically outlawed in the 1789 U.S. Constitution and the Bill of Rights accompanying it.

Thus,

For Quartering large bodies of armed troops among us:

Quartering of troops in private homes is specifically outlawed by the Third Amendment except as provided by law.

He has made Judges dependent on his Will alone, for the tenure of their offices, and the amount and payment of their salaries.

Article III, section 1 specifically mandates that judges serve "during good behavior," and their salaries cannot be reduced as a matter of law.

He has affected to render the Military independent of and superior to the Civil power.

Article I, section 8 places the military under civilian control by giving Congress the sole power to declare war, raise armies and navies, to suppress insurrections and to regulate the militia.

So, as you see, the problem wasn't the law itself. The problem was the King.

In this vein, every state in the Union (Louisiana excepted) has a "reception statute" which imports the common law. I've provided a couple examples below.

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From the North Carolina General Statutes: § 4-1. Common law declared to be in force. All such parts of the common law as were heretofore in force and use within this State, or so much of the common law as is not destructive of, or repugnant to, or inconsistent with, the freedom and independence of this State and the form of government therein established, and which has not been otherwise provided for in whole or in part, not abrogated, repealed, or become obsolete, are hereby declared to be in full force within this State.

From the California Civil Code: 22.2. The common law of England, so far as it is not repugnant to or inconsistent with the Constitution of the United States, or the Constitution or laws of this State, is the rule of decision in all the courts of this State.

Sorry, I know you're probably fed up with Brits saying that you shouldn't say "English" when you mean "British", but in this case it's the other way around. The legal system that the USA, Canada, Australia etc. adopted is Common Law, which is the legal system of England and Wales. Scots Law and Northern Irish Law are different. I know almost nothing about Northern Irish Law. Scots Law has some obvious differences from Common Law such as a different size of jury (15) and a third verdict in criminal cases (Not Proven), but the underlying differences are apparently marked enough that it's not possible to train in one system and practice in another.

This involves quite the explanation. America first adopted a loose confederacy between the states, following her independence. Many states wanted to retain their sovereignty and believed a national government would only be useful for mostly war. However, problems arose during and after the American Revolution. A sharp depression, lack of trading, and states working against one another, rather than together. In 1787 the Founding Fathers wrote the US Constitution due to the problems hurting the nation, such as the inability of the federal government to make money via taxation. They turned to the British form because they knew how stable their government had been, despite recent actions. The separation of powers was an idea taken from the French philosopher, Baron de Montesquieu, who believed if powers were divided, there would be less of a chance of abuse. The checks and balances were written into the Constitution for the same reasons. So, in short, the Constitution included the British style of government because the confederacy failed horribly and they knew it was stable. They changed it from a parliamentary system to allow citizens more power over the head of state through electing them directly (more or less). Hope I was helpful!