In the event of a successful attack on Egypt, then who is going to provide the troop strength to maintain Christian control over Egypt? The country is large, factious, and doesn't have any easily turned minorities. I know the Coptic Christians are there, but they have a good relationship with the Muslims, and would be much more likely to side with the Muslims over the Greek or Latin invaders.
I would have to say that the Latins would be the ones to come out on top in Egypt. With the fall of Egypt the Italian city states will start seeing some really big possibilities of creating some very lucrative trade routes. With bases in Egypt the city-states would have access to the Red Sea and thus to the Arabian Sea and the Indian Ocean. The possibility exists that they could reach out to India, establish there own trading colonies there, and significantly reduce their costs of transporting eastern luxury goods to the West. Basically, with Christian control of Egypt the Italian city states would be able to get direct control of the supply chain that provides them with eastern luxury goods.
Here is what I'm thinking the timeline looks like. The Greeks and Outremer Latins are able to successfully cooperate in '68. They take some fortresses, and invade the Delta. Alexandria falls to a combined Greek-Latin operation, and the force travels down the Nile, attacking Cairo, and briefly occupying the city. They are unable to control Cairo, however, the chaos created by the success of these attacks allows that the Latins and Greeks to solidify control over the Delta.
From their fortresses in the Delta the Latins and Greeks engage in battles against the Fatimids who control Upper Egypt. The Fatimids, facing the threat of Christian takeover, seek an alliance with Nur al-Din, who sends Shirkuh (Salaldin's uncle) with an army to assist the Fatimids. Shirkuh (and then Saladin when Shirkuh dies) is able to gain control of the Fatimids, and push the Christians back until they only control a few coastal fortresses and Alexandria.
Then the Normans come. In OTL 1174 William II attempted an invasion of Egypt, but it failed. Here the operation is a bail-out of the Christian position in Egypt, and with the massive Norman presence the Norman are the ones clearly in control. From the Christian bases in Delta the Normans launch a very successful attack, and in a series of victories (including one where Saladin dies) they push the Fatimids back. Norman troops occupy Cairo, and when disagreements arise between the Normans and Greeks the Normans expel the Greeks from Egypt. William, an ally of the Papacy, gets the Kingdom of Aegyptus (the Roman name of the province) granted to him, and travels to Alexandria to get crowned by the new Latin Patriarch of that city. Egypt becomes a major trading center, with all of the merchant cities having trade colonies in Alexandria and in ports on the Red Sea. Royal taxes on trade goods traveling in Egyptian territory provide huge amounts of revenue for the Sicilian throne.
The real fun starts when Constance and her husband Emperor Henry IV (of the Holy Roman Empire) inherit the Sicilian and Egyptian thrones on William's death in 1189. Their son, Emperor Frederick II, would really love the fact that he has basically has a strangle hold on most Eastern trade. The lever for control of Lombardy may finally have been produced . . .