Whatever you think happens has to be pretty speculative because the game doesn't really comment on much about the societal situation at the time. But that can make it more interesting though IMO.
Here is what I think would happen:
This is assuming several background conditions, that I have deduced throughout playing the FS1 and FS2 campaign many times
- that earth was somewhat if not entirely dependent on the colonies for food and natural resources
- that the GTVI was present in earth's ranks, if not more so than average
- there were a great many military vessels stuck in earthspace due to the impending battle with the Lucifer
- Earth was the center for most Terran R&D as well as manufacturing
- Potentially Earth could find replacement sources for the resources that were formerly provided by the colonies within their own system, they would just be harder to get and take longer to utilize
- People of Earth outnumbered or came very close to outnumbering people in the colonies all put together
I believe that right after the Lucifer was destroyed there would be massive celebrations. However that celebration would be short lived because of of impending supply shortages. As soon as the notion set in that they were trapped in the sol system chaos would ensue.
I think that there would be a power struggle and elements that were a part of the GTVI would come out on top, putting themselves in leadership positions all over. The recovery for earth would be slow at first, but like all humans do they would adapt to the environment. I think that all of this would take somewhere around 3-7 years for earth to be in a stable state again. After this happens earth resumes research and production of new technologies and weapons because for all they know the war is still going on and the shivans could return at any moment.
10-20 years later the reality would set in that the war was most likely over or all life outside of earth has been destroyed. Then they realize that they CANNOT leave the sol system unless they do something to fix their jump node. Sol then begins a new gate project (parallel to the GTVA's.) I believe that the the people of earth would easily be able to come up with a technology that would make that possible. They would presumably have more manpower and their infrastructure was still intact and not touched by the Great War.
After that it would be anyone's guess as to who would finish the jump gate first. It doesnt matter either way. I believe the first encounter with the GTVA would be a friendly one and no one would really want to attack the other. Inevitably tensions would rub everyone the wrong way and there would be some kind of conflict, most likely a crusade against the vasudans.
I think that technologically the earth would easily have the upper hand and would overtake a wounded GTVA if it came to that. After all, earth had pieces of the Lucifer.
Thats my 2 cents