Personally, I think Volt_Cruelerz's idea makes sense, however, lets be honest, it is a bit ambitious to think that all 3 races would make an alliance. I can see, say, 2 out of 3 races making an alliance, but all 3 coming together in a big team up is pretty far stretched.
I'd say that you're pretty close to what would actually happen, however, neither TR or TL would be a part of the alliance. Both factions are self serving. I'd bet a ton of money that the TL would either be too afraid/resentful to let other races into their plans, and, along with trying to isolate themselves, would try to keep anybody on the outside from coming in. They're not going to want to share their plans or resources, or even the trouble of looking after someone else. I'd say they're just concerned about the survival of the human race, and they are willing to leave the other races, the Rebels, and the Neutral factions out in the cold until they've all killed each other off.
I can see the Vasari and Advent becoming allies- they actually don't have problems with each other, as far as I know. (actually, for that matter, I seriously doubt that the Vasari are really resentful about much at all) The Advent could probably swing either way as to their stance with the Vasari, who, if given enough incentive, would probably be willing to include others in their plans.
I'd say that the only 2 ways for the 3 races to stand together, would be if the above alliance between any of the Vasari or Advent factions had TEC forces under their control- The Vasari's tech tree enables them to force a planet's inhabitants into slavery, and the Advent can convert TEC (and Vasari too) to their cause. The other possibility is that while I can't see the TR/L joining this alliance, there are other human forces out there. As in, the Pirates and Neutral factions. Provided their story is different from the TR/L, I'd say them joining this big alliance is always possible.
Well, I'll say again that your story is solid, makes for good lore, and would make a great backdrop to future installments (or even, gasp, maybe an ambitious modder...?) I'd say it's just far fetched, and I wouldn't bet on it actually happening, in real life. Not to say that Sins isn't just a game, or that that alone is reason why your story shouldn't happen, of course.