When America declared its independence from Britain, it took many years of convincing the nation to unify behind a single government. However, when the CSA declared its independence from The Union, it took just a few months to ratify a constitution and a year to create a government with legislature and president that saw legitimacy from most southerners. How did The Confederacy create a government so much quicker than The United States?
So your question actually makes an incorrect assumption. The federal government created from the United States Constitution in 1787 was not the first government that ran the combined 13 colonies. Instead, the first government that was created was under The Articles of Confederation that was drafted in 1776, the same period that the Declaration of Independence was drafted. The first draft was presented to the Continental Congress on July 12th, 1776, mere days after the Declaration of Independence was created and endorsed by congress.
The Articles of Confederation was an interesting government. While it was created in July 1777, it went to the states to be ratified. It became the governing model for the United States government in 1777 and was finally ratified when Maryland, the final colony, passed it in 1781. It then existed legally a the US's system of government until the Constitution replaced it.
The A.O.C. provided for a unicameral legislature without a judiciary and without an executive. The government was relatively weak, but it did allow the for the Congress to have some important powers. It could:
However, it had some pretty big drawbacks. It could not:
The real challenge was that last point. Each state could have between 1 - 7 representatives in the Congress, but they only received one vote. Every single vote had to be passed by all states, which was an insanely difficult. This was exacerbated by the fact that the 1780s was a very difficult time in America. It was experiencing an economic crisis unlike anything the young country had ever seen before. Leaders across the country couldn't get anything passed with the help of the government because it was too defective, so things needed to change.
The deficiencies of the Articles of Confederation led the Constitutional Convention of 1787. In less than 4 months, the 55 representatives who came from the colonies drafted an entirely new form of government almost from scratch. This was after countless debates, long dialogues and monologues, 39 delegates signed the document when it was completed. I mention this because your initial question presupposed that the Confederacy's constitution was drafted quickly, which isn't entirely true. Especially since it is nearly a word for word copy of the United States Constitution with a few key changes. This means that the 1787 United States government drafted a whole new constitution in nearly the same amount of time that it took the Confederacy to essentially copy the same document while making a few changes to support their views on slavery and a few other issues. They also had the advantage of knowing how a democratic republic should work, where as their 1787 counter parts were, essentially, making it up as they went along.
Edit: clarification around one point