Clearly, British laws against e.g. murder, theft, fraud, and most
other crimes were received into state law because those matters are
under the jurisdiction of the states.
States (and also to some extent federal criminal statutes) received British common law definitions of various crimes and defenses, but not British penal statutes (which often didn't define those crimes in the late 18th century).
The statutory law of Britain did not apply as U.S. law at either the state or federal level upon the U.S. Declaration of Independence, except in isolated cases where a state, or the federal government expressly adopted it by reference in their own statute.
The existence of British law, both statutory and through case law, informs how the U.S. common law was understood (something that was predominantly a matter of state law) and how concept in the U.S. Constitution, U.S. federal statutes, and state constitutions and statutes were understood (especially when terms from British statutes are used in a similar matter in U.S. state and federal statutes and constitutions). But, British statutes did not have direct force and effect in the U.S. after independence.
As much as anything else, this simply reflect how the nature of statutes v. common law was understood in the late 18th and early 19th century. Nobody expected that British statutes would be directly applicable, so they weren't.
There may have been instances where common law rules actually had their roots in British statutes that were mostly forgotten in long layers of British common law case law, and many statutes expressly adopting British case law also expressly incorporate selected relevant British statutes of generally applicability. But, no British statutes were applicable "automatically" in the U.S.
For example, before Congress enacted the Naturalization Act of 1790,
would a person born outside the US to a US citizen father have been a
US citizen by dint of the US "inheriting" the British Nationality Act
1772 mutatis mutandis?
No. Basic ways of thinking about what nationality or citizenship even was or meant would have been received, but not by receiving the British Nationality Act 1772 as U.S. law.
And before Congress enacted the Crimes Act of 1790, would an American
who levied war against the US, gave aid and comfort to its enemies, or
counterfeited US currency have been convicted and sentenced under the
Treason Act 1351?
Not really.
From the adoption of the Declaration of Independence on July 4, 1776 until the adoption of the Articles of Confederation on June 11, 1777, as a practical matter, the revolutionary forced had very little actual control of the courts in an enforceable way, there was an insurgency against the British underway, and it was an ad hoc effort from day to day and month to month that was muddled through without a formal structure or guidance at a colony by colony level, or even more granularly within a colony.
In the period from June 11, 1777 until the new U.S. Constitution was implemented in 1789-1790, under the Articles of Confederation, the situation was fluid and irregular. State governments made most statutory and common law, adopting their colonial era colonial statutes but not necessarily British statutes that their local legislature didn't adopt. Central government laws usually acted on the states, not directly on individuals, much like treaties today. There was little or no directly applicable central government legislation.
Many topics, like citizenship, were simply ignored in this era. The U.S. Constitution adopted in 1789, and the initial acts of the First Congress to implement it, were a response to the realization that after the war and its immediate aftermath had settled down, that the newly formed country needed to regularize, institutionalize, and deal with a lot of governance issues and legal questions that nobody had had the time or resources or authority to deal with while a war had been going on.
For much of this time period the Revolutionary War was in progress and it wasn't always obvious who even controlled the courts or had practice authority to enforce court judgments. The Revolutionary War was not concluded until 1783.
It was an improvisation at first, and not necessarily a uniform one, since the Articles of Confederation conceived of the U.S. as many countries in an alliance with each other rather than an actual single nation that had to address legal issues uniformly.
Prior to the establishment of the federal court system under the U.S. Constitution of 1789 that remains in force, the only institution of the central government was Congress and its committees, which functioned as a legislative body, a body selecting people with executive authority, and as a court of last resort from state court judgments. Everything was carried out at the state level except for courts-martial.
Structurally, the Articles of Confederation were a fused system, akin to the U.K. Parliament which had its highest court of appeals and its prime ministership fused with the legislative authority of parliament, layered on top of 13 separate sovereign state governments.
