One thing I think would happen is a possible Norman civil war between Robert of Normandy and King William II of England, since the Crusade wouldn't carry Robert off, he wouldn't mortgage Normandy off to his brother, and with all his lands, resources, and allies, would probably look to take all the Norman lands for himself. Their brother Henry may never get the real opportunity to rise to the throne himself if his brothers are fighting over it, but probably somehow finds himself on the winning side (he was shrewd like that) and gets himself some lands and a prestigious marriage. If he doesn't rise to the throne, that means no Empress Matilda, no Anarchy, and no Plantaganets. The entire course of English history is completely butterflied away. I wish I was more familiar with the other leading barons to better determine what their likely roles would have been without the Crusade.
At best, Alexios Komnenos perhaps doesn't send a desperate distress letter to the Pope, but individually contacts reliable barons like Robert of Flanders, who had fought for Alexios at one point. I wonder if he would be desperate enough to hire Russian mercenaries, who would be recently converted themselves to Orthodox Christianity (relatively speaking, of course, Vladimir of Kiev converted in 988). I have to imagine Orthodox "crusaders" in the employ of Constantinople would be easier to handle.