United Earth was first established (chronologically) to exist as Earth's unifying world government in "Home" (in which we saw the UE seal for the first time) and again in "The Forge," where its name was clearly established on the sign of the United Earth Embassy. "Demons" established that very high-ranking members of the United Earth government receive the title of "Minister," implying a parliamentary government but not explicitly establishing it.
The novels have established more. In the ENT novels The Good That Men Do and Kobayashi Maru by Michael A. Martin and Andy Mangels, Nathan Samuels is established to be the Prime Minister of United Earth, and the UE PM is established to be the head of government. The UE PM is confirmed to exist as late as 2375, when the Prime Minister of United Earth joins the President of the United Federation of Planets and President of the United States of America in touring the devastated San Francisco after the Breen attack in the short story "Eleven Hours Out" in the anthology Tales of the Dominion War." The Corps of Engineers novel The Future Begins establishes that United Earth was also led by a President by the name of Lydia Littlejohn who was remembered for her inspiration of the populace during the Earth-Romulan War. The Parliament of United Earth is established as existing in A Less Perfect Union, a Myriad Universes novel whose turning point comes in ENT Season Four. Kobayashi Maru establishes the existence of something called the United Earth Council.
If we take the novels into account, all this implies that United Earth is a parliamentary republic, with a President of United Earth who serves as a mostly-ceremonial head of state, a Prime Minister who has real power and serves as head of government, and a Parliament that serves as the legislature. Exactly what the United Earth Council is isn't established in Kobayashi Maru, but I would theorize it to either be the official name of the cabinet, the name of the upper house of the Parliament, or another name for the United Earth Parliament.
The Good That Men Do refers to United Earth as being a federated government, implying that it practices federalism -- the division of authority between the central government and regional governments. As such, I'm sure that each member nation of United Earth gets to pick its own government and has its own areas of authority, provided that they're all liberal democracies. (I can't imagine that dictatorships are okay on UE.) We know from ENT Season One that Malcom Reed's dad expected him to join the Royal Navy like the rest of his family, which implies that the Royal Navy of the United Kingdom still exists. I would take this to mean that United Earth utilizes the armed forces of its member nations for some Earth-bound functions, and that the Royal Family of the United Kingdom still exists and still ceremonially reigns.
As for Vulcan, "These Are the Voyages..." established that after the dissolution of the High Command and the overthrow of Administrator V'Las, the Vulcan legislature was called the Vulcan Council. "Home" established that Vulcan was headed by a First Minister before the P'Jem incident in "The Andorian Incident," and that after this the First Minister was forced to resign and V'Las assumed power -- a military coup. Kobayashi Maru established that the new civilian head of government after V'Las was overthrown was once again titled First Minister, and that the Vulcan state was now known as the Confederacy of Vulcan. The Good That Men Do and Kobayashi Maru establish the First Minister of the Confederacy of Vulcan to be T'Pau -- which makes sense, since it was T'Pau that T'Pol appealed to for more Vulcan ships in "United."
Other novels have established that Federation Member States get to send one Federation Councillor to represent them on the Federation Council, and Articles of the Federation establishes United Earth's Federation Councillor to be a man named Matthew Mazibuko from Africa, whilst the Confederacy of Vulcan's is a woman name T'Latrek. There is no evidence from any source that the Federation directly governs Earth -- indeed, if it did, it would hardly be a federation (meaning a state that divides authority between the central and regional governments), now would it?