Alright, here goes:
The Saljuq Khanate
What happened to Baiju, he who brought Rum and Anatolia under Mongol control?
After the battle of Kose Dagh and his preoccupation in the Rum area, Baiju discovered that Azerbaijani and Ayyubid forces had cut him off from Persia. Receiving news that Persia had thrown off the Mongol yoke, and observing the battle weariness of his troops, he chose to camp down, and ordered Chormaghun to do the same.
All around him, hostile enemies stood: Azerbaijan to the north, Ayyubids to the south, Byzantium to the west, and the Caliphate to the east. The last Crusader states (i.e. Aleppo) made overtures to the Mongols, which were accepted, but shortlived with their destruction by the Ayyubids. Things got easier when first the Caliphate fell to a Shia force, then the Ayyubids fell in Syria. But these enemies were replaced by Azerbaijan and the Mamluks. Luckily for the Mongols, these two kingdoms fought each other for control of Iraq. Byzantium was far too busy in the Balkans to hurt the Saljuqs seriously.
It was mainly a game of survival for the Saljuqs. In the early 14th century, the Ottomans were supressed, and absorbed into the armies of the khanate. Sunni islam was adopted by the Mongols, and moves were soon made against the Christian heathens in the area. This forced the creation of the Byzantium-Venetian alliance, as both countries were eager to protect their interests in the Levant.
The invasion of Timur weakened the Azerbaijanis and the Mamluks. Saljuqia would have been next, were it not for Timur swerving north to fight the Golden Horde in Azerbaijan. So, with their eastern enemies down and out, the Saljuqs could wage war against either Byzantium or the Mamluks...
What thi Changes from what I had Before:
-Ottomans do not exist, and neither do the smaller Turkish sultnates.
-Byzantium still has a firm hold on western Anatolia
-The invasions of Greece and Varna by the Ottomans don't happen, so the reason for Varna joining the Christian alliance are removed. Instead, tp help them deal with Wallachian and Moldavian revolts, Varna invites Byzantium as a friend, entering her into the allaince with Byzantium and Venice (Balkan League). This of course increased discontent amongst the peoples, who still remembered the destruction caused by Byzantium in the Great Euro-Mongol War.
-Krementjug does not have a friend in the Ottomans anymore. Until the loss of Varna in the First Great Euro-Mongol War (when that area was still Krementjug), a friendship had been built with their Mongol friends across the Black Sea. Trade developed, and even a secret alliance. Krementjug's funding of Saljuq wars against the Balkan League did not endear them to their southern neighbours in Varna.
-Mamluks are in alliance and vassalship with Timur, not only to preserve themselves from destruction, but to protect themselves from the growing power of the Saljuqs.