I am curious about the exact events that would have taken place in the Commonwealth, seeing as they would be quite significant and would most likely affect the areas in the west also.

In my opinion a victory against the Institute on behalf of the Brotherhood of Steel would give them full and unchallengeable control over the entirety of The Commonwealth. Not only that, but by destroying the feared boogyman, their already massive ranks would see a surge in new recruits. This leaves them in control of both The Capital Wasteland and The Commonwealth, their ranks enlarged quite significantly. 

It is very likely that they would send troops back to California to deal with the NCR, which in my opinion would be a fight in which they would most likely stand victorious. I would not be surprised if at the very least, patrols were send into the Canadian regions in search of technology.