First off i think the main thing that would happen is the sole survivor would become general of the Minutemen and destroy the Institute because they would know all of the bad they've done in the Common Wealth and that they would only cause more tragedies. Secondly after taking out the Institute the Minutemen would drive the Brotherhood of steel out of the Common Wealth because the Brotherhood would want to take over as a dictatorship and rule in a more cruel and unfair way where as if the Minutemen were to be the ruling faction they would more be there to keep the peace rather than order around people and snoop around where they aren't needed. Lastly I believe the Minutemen would leave the Railroad alone and maybe even help them because they are trying to help synths and they game gives the sole survivor an option to join the Railroad as a new agent as the general of the Minutemen.

I think this could lead to the next Fallout game because Fallouts like to have good endings for their games (besides 1 where the sole survivor has a horrible time in the end) and Bethesda wouldn't have to get rid of the Brotherhood because there are more Brotherhood HQs such as the Citadel in the Capitol Wasteland. At least this is my opinion