Let's say instead of Chinggis Khan, a khanate under Jamuqa forms instead. Jamuqa manages to unite the Mongol tribes, and builds a broad Mongol Khanate. However, lacking the aptitude and generalship of Genghis, he fails to make forays into Jin Dynasty. Instead, his khanate edges out an uneasy peace with the Jin with a trade of horses for silks and other fine goods. His political structure does not last beyond his death, however, and the Mongol Khanate based roughly along the borders of modern Mongolia collapses back into internisine warfare and tribal conflicts.
The Jin dynasty lasts for another fifty or so years, before dynastic decline overtakes them and they collapse again. At this point no one could say for sure who would surplant them, possibly another more uncivilized Jurchen, Tungusic, or Mongol tribe from the north, or maybe a north Chinese dynasty. Regardless, China would probably stay disunited.
The Song, untroubled by Mongol attacks would last possibly a bit longer than the Jin, but would ultimately face the same problems as dynastic decline takes its toll. They too would probably fall around 1300 or slightly earlier.
The Khwarezm empire would become the behemoth of the Middle East, and around 1230 would have managed to integrate the Turkic peoples far more thoroughly into the military apparatus. As a result, it would become immensely rich and powerful. Similar to the later timuriuds, a turkic-persianate court culture probably emerges. The Khwarezm dynasty probably would also become a Shia state, and if conflict with Baghdad remains, would probably turn them into a vassal ala the Seljuks. Regardless, around 1350 or so the Khwarezm state would fall. Greater Iran, such as Tajikstan, Uzbekistan, and Turkmenistan would have a stronger Iranian influence and culture rather than Turkic of today.
Egypt would probably stay Ayyubid, or decline later into a native or more likely foreign sultanate dynasty. Mamlukes never become prominent as the Mongol conquests don't happen, or could possibly remain an institution that never gains political power.
Russia remains divided, Kiev and Chernigov remain strong. This probably would last until 1400, when early modern states start to expand. Russia would probably never come into being.
Zen buddhism never quite takes off in Japan due to the lack of the Mongol invasions, and instead Japanese Buddhism remains more similar to previous antecedents.
The ottomans and other anatolian emirates remain under the sultanate of Rum. They don't grow as fast or as powerful RTL.