The US can't start off with a Westminster style system. It's too British.
Our current system's not that far from it. Just separation of the legislature and head of government. However, we have retained the bicameral legislative structure, the popularly-elected lower house based on proportion, and an upper house that was originally a fairly aristocratic institution composed by people appointed by state legislatures. It was only with the 17th Amendment in 1913 that Senators were made to be popularly elected.
The early US was essentially Britain Mk.2; our culture, our government, our legal system, our very notion of liberties and rights afforded to all men, were derived from British tradition and thought. With changes made to incorporate Separation of Powers and Federalism.
So us adopting a Westminster-style parliamentary system in our early years isn't that improbable. Just have the Anglophile merchant class and the Federalists have more say in how the constitution is written. The US would end up awfully similar to what Canada came to look like; albeit with an elected head of state.