While parts of the late Mongol Empire saw the rulers convert to Islam others did not. The first thing to note is that in the Mongol court of Genghis Khan and his immediate sucessors a multiplicity of religions were practiced by the Mongols including, but not limited to, Tenigrism, Buddhism, Islam and Nestorian Christianity. While different religions had more influence than others at various times none was ever dominant due to 1) Mongol policy of tolerance and 2) the fact that the Mongol elite was very religiously diverse (though many high ranking Mongol women were Nestorians).

After the Mongol Civil War three main sucessor states emerged. The Yuan Dynasty that ruled China and most of what we now consider China, the Ilkhanate that ruled what might loosely be termed Greater Iran and the Golden Horde which dominated Central Asia and Rus areas. Here's a map to help clarify. It was only after the Mongol civil war that Mongol courts which were predominantly Islamic began to emerge. The first was the court of the Golden Horde followed by the Ilkhanate in 1295. However, as you see the Yuan court which lay outside the Islamic world did not convert. As you can see they controlled much of the area we now see as Mongolia proper. However, lest I give too simple a picture bear in mind two things 1) there is an Islamic presence in Mongolia of noticeable size (about 5% of the population) and 2) even if the court had converted this would not have meant most of the population that remained in the area that now constitutes Mongolia would have converted. The official region of the Yuan court was Tibetan Buddhist but most the population on the plains probably remained Tenigrist/shamanist.