Dulahan over on RPGnet justly called me out on the way I had glossed over the time between the death of Saladin and the game date of 1560. Specifically, as a Mongolist, he insisted I had to deal with the Mongol invasion under Hulagu and the Tatars under timur (Tamerlane). I had glossed it over because I was in a mental hurry to get from the formative years to the action, sort of like in the gospels where Jesus is a baby then suddenly an adult. Dulahan was correct! I needed the detail those in-between years enough to explain where we are "now".
Hulagu was a grandson of Ghengis Khan, and one of the greatest Mongol kings. He had been assigned by his brother, the Great Khan Mongke, to deal with three problems in the middle east - the Assassins of Persia and Syria, the Mamluks of Egypt, and the Abbasid Caliphate in Baghdad. In the real timeline, he conquered all of Persia, wiping out the headquarters of teh Assassins at Alamut, took and sacked Baghdad and destroyed the Caliphate, then conquered Aleppo, Damascus, and Homs with the help of his allies, the Christian princes of Outremer. He turned back with most of his army because of the death of Mongke, and the need to elect another Great Khan - in this case Kublai Khan. A far smaller force under Kitbuqa was crushed by the Mamluks at Ayn Jalut in Galilee, and the Mongols were never able to sustain further conquest, though their Ilkhanate based in Persia endured until the time of Timur.
As an aside, Hulagu was very friendly to Christians - his mother and wife were Christian, as was his close friend and general Kitbuga. He may have been Christian himself, though this is in some doubt, and there is evidence he may have been Buddhist. The Mongols had complete freedom of religion, and individual Mongols chose their religions for themselves, rather than having it imposed.
In the late 14th century, Timur swept through the region. Timur was a descendent of Ghengis Khan, but his people were a mishmash of Mongol and Turkic people called the Tatars. He claimed legitimacy as a Khan, but he was never given it - he had to take it. Timur and all his followers were Muslim, though he killed far more Mulims than infidels. His Tatars destroyed the Ilkhanate, razed Baghdad with extreme prejudice - about 9 out of 10 inhabitants were killed, took Syria, then hammered the Ottomans - who took over a decade to recover - and Mamluk Egypt. He then turned his attentions to the east, and in the first years of the new century died on the verge of conquering Ming China with a vast horde.
In the Outremer timeline:
In 1192, after marrying his brother to Joan of Sicily and forming the new Muslim Kingdom of Jerusalem, Saladin was assassinated in Damascus. Simultaneously, the Assassins took Homs by treachery, and moved many of their people there over the next few years. Saladin never took anything beyond Syria, and Egypt remained under the Fatimids, who were not enemies of the Mongols, unlike the Mamluks.
In 1260, the Mongols under Hulagu allied with the Prince of Antioch and the Count of Tripoli, smashed the Assassins in Masyaf, taking Damascus, Homs, and Aleppo as well. Jerusalem offered tribute to Hulagu before it too was crushed, and Hulagu left, his mission completed, back to Persia.
By this time, though, the ruling clans in Homs and Hamah were decidedly Ismaili, and after a succession of Ilkhan governors, regained their independence after Timur rolled through everything to the east about 1400, along with the Sunni powers in Aleppo and Damascus.
There was no campaign in Egypt, the Mamluks never took power as the slaves from the north went elsewhere. The Fatimid Caliphs in Cairo were dissolute and nowhere near as warlike as the Mamluks. They made token submission to Hulagu, and were left in place.
Baghdad was revived from near extinction after the sacks of Hulagu and Timur, when remnants of the Ayyubids regained control of Iraq in the late fifteenth century. It still has nowhere near the power it once had, but controls most of the Tigris and Euphrates valleys.
Timur had two goals in the real timeline, crushing the Mamluks and the Ottomans. There were no Mamluks, so he turned on the Ottomans alone, staggering them for decades, before turning back to the East. His descendants - the Timurids - never amounted to anything, and his empire dissipated.
This gives me what I need for the setting, and actually explains the survival of the crusader states better than what I had. Being allies of Hulagu - which they were in our timeline as well - would leave them more strongly in place. The Ottomans were not destroyed, but were hit very hard, and took long to recover - leaving Armenia and Edessa in place, and allowing for Constantinople to survive.