My guess is a mini-GTVA would have developed. Eventually.
Since there were some Vasudans in Sol, my guess is the initial xenophobia and panic after the collapse would have forced the Vasudans into the outer areas. Neptune, Pluto, Uranus, etc. since the humans didn't quite want to mingle so freely.
Earth, Mars, and Venus all go independent, then they quickly unify under a federalist government with a large degree of planetary autonomy similar to the United States, Ceres quickly joins since they are scared of the Jovians and Saturnians, who have unified into a semi-fascist government, and the Vasudans are setting up in the outer reaches, mining the Kuiper Belt, coexisting peacefully with the humans there, and they form their own government similar to the United Kingdom in structure and distribution of power, just no hereditary monarchs (Vasudans like Parliaments, and the humans out there probably hate any sort of hereditary leadership, kings, emperors, and nobles so a compromise is struck). And trade quickly springs up between the Earth-based and the Pluto-based governments once they realize the other exists, and they get along well. The Jovians and the Saturnians are not amused, and decide take the solar system by force. War ensues, they are defeated, and the Vasudans and Terrans decide to make the bonds forged in that war even stronger by completely unifying under the banner of the Galactic Terran-Vasudan Alliance.
And serious problems ensue when the Delta Serpentis-Sol node is opened, since there can be only one Galactic Terran-Vasudan Alliance. And the Sol one is set up quite differently than the Beta Aquilae one since it is set up more like the United States where the Beta Aquilae GTVA has more in common with the U.N.