The US used to be a British colony, yet unlike lots of other ex colonies it doesn’t follow the British parliamentary model of goverment. Who came up with the US model of senate, congress etc.
When was the design finalised (I realise that like all democracies still evolving)? i.e. how long after the Declaration of Independence did Americans have to wait to get a government that would be recognisable today?
The US struggled along under a government organized under the Articles of Confederation for something like 16 years after the end of the Revolution. By the late 1780’s it was decided to do something because the central government was too weak to enforce taxes, import duties or national laws in general and it hurt our internation credit and the conduct of trade. So a group was named to modify the Articles. Instead they wrote and entirely new constitution which was adopted in 1789 and that’s the form of government in use today.
How was the government under the Articles of Confederation organised?
Someone who knows for sure will probably be along shortly. I’ve cited a copy of the Articles and there are a lot of on-line sources. Search Articles of Confederation.
It was a relatively loose confederation of relatively independent states.
No one really came up with it. It was the result of compromise. The bicameral legislature recognizes the power of the states and the power of the majority
It’s true that the current system was established with the Constitution and remains essentially unchanged.
But it’s also true that the U.S. never at any time had a British-style parliament or prime minister. A Congress was already in place at the time of the Declaration of Independence. This Congress was a collection of representatives from each of the colonies, although they were appointed rather than citizen-elected.
The idea of the colonies, or nascent states, being the core division is already a very different notion than the British division into boroughs or ridings. It’s this underlying reality of virtually autonomous separate areas coming together to build a system from the bottom up that creates the fundamental distinction to the British system, which was top-down in all ways that mattered.
The Constitution codifies this, but is basically just a better and more workable political compromise from the Articles of Confederation or the earlier ad hoc Congresses than a qualitative change in government.
I think part of the answer as to why the US did not adopt the Westminster system is that it became independent before the Westminster system was adopted in Westminster. Back in the late 18th century, the English monarch really did govern, though he or she did have to get certain things through Parliament, like supply bills. It was in the 19th century that the monarch became a figurehead. The British changed that because the Prime Minister, owing his position to support from the electorate, had the moral and political authority that a hereditary monarch did not. On the other hand, the US had an elected monarch in the president, so the president retained and expanded on the powers of the British monarch (subject, again, to what he could get through Congress).
The other thing is that the writers of the US Constitution were concerned about separation of powers between the three branches of government. This has never been much of an issue with the British constitution: the executive and the legislature are intertwined, with the Prime Minister at the head of both. (The British judiciary is more independent, though you still have the House of Lords as a body exercising both legislative and judicial functions).
No cite, but I once read that the U.S. Constitution was roughly modeled on the state constitution of New York. But that raises the question of why New York had a separation-of-powers system instead of a parliamentary system. Why none of the colonies ever developed parliamentary government, I cannot say – probably it was because there was an elected legislature and an appointed royal governor in every colony, and the idea of a colonial legislature putting up its own “prime minister” might have seemed like a treasonous challenge to the governor’s authority; and then, after independence, the states simply stuck with that model, but with an elected governor.
Whatever actually inspired the separation-of-powers system, theoretical justification for it was found in the political writings of Baron Montesquieu. The Federalist Papers cites Montesquieu and asserts that for the creator of the laws to also be their enforcer is "the very definition of “tyranny”. Which might have seemed plausible at the time, looking at Britain which was – then – effectively a tyranny, not of the king, but of the landed gentry that controlled Parliament.
You can find that discussion, by the way, in The Federalist No. 51.
I believe the State of New York may have been influenced by the system employed by the Iroquois nation, a federation of 5 tribes mostly living in New York State, formed in the 1600s. Several of the founding fathers (Benjamin Franklin for one) were great observers of the Iroquois system.
The government of the United States was organized by the Constitutional Convention in 1787 and enacted in 1789. The primary basis of the United States government prior to this had been the States, which were organized into a loose national structure by the Articles of Confederation. Many Americans realized there was a need for a stronger national government if the country was going to be stable (or even survive). So the Constitution was designed as a compromise that would produce a strong enough national government to rule the country but weak enough so the states would retain some power.
Inspirations for the Constitution came from a variety of sources including existing state constitutions, British law, classical forms of government, and political theories.
I am under the impression that the basic structure of the Constitution derives from The Virginia Plan. The delegation from Virginia arrived at the Constitutional Convention with this in hand and no other delegation had anything comparable.
Well, you can’t fight something with nothing so the Virginia Plan was a point of departure and formed the basic framework which was modified as the Convention proceeded.
Ah, but between the Virginia plan (for representation of states in proportion to their population) and the New Jersey plan (equal representation for each state) came the wisdom of the Connecticut compromise, with one house each way.
As to the OP, the constitutional convention hammered out pretty much all of the outstanding issues (representation in government and taxation, role and power of branches of government, powers and limitations of government, the western borders of the states (large areas west of the original 13 colonies were claimed by more than one state), how to add new states, etc.) In some cases the compromises were such that all of the signatories and states would agree on the meanings. On the rest things were left worded such that all sides agreed on the words if not the meaning and interpretation.
It’s also somewhat important to know that while the shape of the government was mostly set out by the Constitution, certain aspects of it, most importantly judicial review, evolved over time and precedent. Judicial review is not in the Constitution - the precedent for it was established by Marbury v. Madison in 1803.
A perceptive foreigner might notice by that that the Supreme Court itself essentially decided what its role was supposed to be. 
I found a site that gives the details on every plan the Convention considered – http://www.usconstitution.net/consttop_ccon.html:
There was considerable discussion of the roots of the constitution here. It contains copious references - you might even call it a bibliography. 