OK, I just received my copy of GURPS Infinite Worlds, and reading through it, I came across an interesting 'pocket scenario' where the Mongols ravage most of Europe. Lots of refugess flee to the Crusader states, giving them a big demographic jump. The Crusaders then ally with the Ilkhans against the Muslims, allowing Egypt to become Crusader Egypt.
It's a fascinating idea, and I think I'll do a TL on it after Washington's Warlocks is done. But for now, here's an idea of what the middle east might look like: