The Mongol invasions of India were halfhearted in the beginning and done by remnants and shadows afterwards, so they are not reliable indicators of what could or could not have been. I don't put too much faith in their ability to conquer India, rather I'd put it with just about as much faith as one would have for Southern China falling. Southern China would be conquered by the Chinese from the northern states and state of the art technology, India might be conquered in a similar way. It might be interesting to see what would happen if India fell, but the Song endured. In fact that might be a very good premise for a timeline.
As to the topic of the religion of the Mongol invaders, should it occur while the Horde is still the Horde and not only a dejure federation or even worse a remnant of a remnant. If its in the beginning there's a very good chance that the Mongols will adopt the religion of the bureaucracy and ministers inside of India. Given the rather intense hatred towards Muslims (at least at the start) the taking of India could provide a shift away from the Mongols even becoming predominately Muslim. The conversion of the steppe to Islam being a product of the Mongol invasion. This also might prevent the rise of a genocidal madman who butchered the Middle East in such a way that a primarily Christian region became a predominately Muslim one built upon the foundation of tens of thousands of corpses. Which in turn has its own little butterflies.
Should it come later, particularly after the Il Khanate converted to Islam, then there's no chance of a Hindu Mongol Empire. At least, not within reasonable confines of projection. You'd need to force it to happen.