If it were the entire Horde vs. the entire Alliance in the middle of Org, I could see the Horde chasing the Alliance out. But this is half of the Horde Vs. half of the Horde Vs. the Alliance. The two Horde forces will be thrown into disarray. There will be chaos. Horde has never fought Horde before. But the Alliance? This is what they do all day, every day. They'll be organized and have numbers.
If the Alliance has 100 people at the siege and the Horde has 100 but is split 50/50 because of Garrosh Vs. the Rebellion, if Garrosh loses and both the Rebellion and Alliance take 50% losses, that still leaves the Alliance with 50 and the Rebellion with 25 giving the Alliance a nice edge. And that's being generous and assuming the Alliance lose 50 and the Rebellion loses 25.
The numbers just don't make enough sense for the Rebellion to force out the Alliance.
The numbers have a few other problems than that though.
It's not "the entire Alliance" vs "just the Horde rebels". It's all the forces the Alliance can pull out from every other front on the planet in time for the Siege vs all the forces Garrosh can pull out from every other front on the planet in time for the Siege and the Horde rebels (who're not operating on any Alliance Horde front, so they've presumably got a greater ratio of participants and total members)..
Then there's the fact the Horde, Garrosh and rebels both, have home advantage and the backing of three nearby cities (downplay it all you want, three cities to the Alliance's zero is
Plus the fact the Alliance lost the top tiers of leadership of just about every single one of its militaries when they all showed up in time for Theramore to explode, so it's unlikely their advantage in sheer numbers is that huge anyhow.
There simply isn't any basis to be going around saying the Alliance's presence in the Siege of Orgrimmar means they've totally conquered the city and that there's nothing anyone can do about it. Sure, there's similarly nothing saying the Horde could just kick the Alliance out afterwards either (except for metagame, but I'm traipsing around that for now), but hey.