And indeed, there were two Presidents who were born after the United States came into existance, but before the Consittution was ratified. Presidents Martin Van Buren, born in December of 1782 and Zachary Taylor, born in November 1784.
But the period was not really all that long. July of 1776 (at the earliest, some might date the existance from ratification of the Ariticles of Confederation (March 1781), or the Treaty of Paris(signed September 1783, ratified in January 1784) to June 21, 1788 when the Constitution was ratified. I used the 1776 date, but you'd get the same result if you used the 1781 date of the Articles. I would not use the Treaty of Paris, since that was only getting the Brits to aknowledge what had already been created.
But it really doesn't matter because both were Natural Born Citizens of the United States and were also citizens at the time of the Adoption of the Constitution, as was everyone then living in the United States that did not choose to evacuate to Canda or England.
But other than those two, who could hardly be considered founders, the rest of the first 12 Presidents, where not natural born citizens of The United States. They were natural born subjects of King George of England, Although all were born in the King's American colonies, it would have been possible for an immigrant from either another British colony, like Alexander Hamilton who was indeed a founder, to have become President during that period, as long as he was living in the US at the time of the adoption of the Constitution. So could Friedrich von Stueban,a Prussian by birth, but who was the first Drillmaster of what was then the Continental Army. He became a citizen in 1783, before adoption of the Constitutioin. (But virtually no one wanted him to be President, probably not even himself)
Regarding Friedrich von Stueban...it’s amazing what he did to help the troops, especially while at Valley Forge!