The question is when
The main question you need to answer here is when the Mongols are going to conquer India. There are three opportunities that can cause Mongols to conquer India.
1) When Jalaluddin (son of Khwarazim Shah) was fleeing from the Mongols (approx. 1221). he fled towards the Indus and sought refuge from Iltumish. Iltumish denied him refuge and Jalaluddin fled to Sindh, plundering his way across the Punjabi countryside. But, here let us assume that Iltumish offers refuge to Jalaluddin. Genghis Khan was the one pursuing Jalaluddin, so even if he did not immediately conquer, he would, sooner or later, crush Iltumish and Jalaluddin. If this happens, then the Muslim power in India is doomed. Muslims had barely been ruling east of the Ravi for 25 years by this time, and a large number of small, but independent Hindu kingdoms were still around in northern India. The small number of Muslims who formed the elite of the Delhi sultanate would likely be butchered when Delhi fell (or when they fell fighting in battle) and there is not much chance of the Mongols turning to Islam, if for no other reason that there is not much Islam still left around in north India. In this case, I can see your scenario unfolding, with the Mongols turning to Hinduism, and they being absorbed into the cultural fabric of India. Further, Muslim power in India at this point was weak. It was the Mongol invasions that strengthened the Muslim power in northern India, since it sent several waves of tribes who had been uprooted in central Asia to flee to the Delhi sultanate for refuge. The Delhi sultanate happily accorded them that, and they strengthened the Muslim power in northern India.
2) The second chance came during Mangu Khan's advances (~1255). There was a debate about whether they should proceed towards Baghdad or whether they should advance into India. Ultimately, they went west. But if they went against India instead, they may find that Delhi sultanate is about as powerful as Mameluke Egypt. Still, the Delhi sultanate was divided at the time, and is not likely to be able to stand up against the full power of the Mongol army, and they are likely to be crushed. In the 4-5 years of campaigning that Mangu Khan had, it is likely that the Delhi sultanate power, all the way up to Bengal, is going to be crushed. However, there would be a Muslim elite still around in north India. The Mongol conversion to Hinduism would come about if they decide that it is in their interest, and there is a powerful Muslim power to the west that requires the Mongols to look for non-Muslim army/bureaucracy recruits. This is possible.
3) The last chance was in the 1290s. A large number of Mongols were plundering Punjab in confused squabbles with Zafar Khan (a general of Alauddin Khilji). If the Mongols are victorious, they are still likely to convert to Islam, since the bulk of the elite in northern India was Muslim at the time. There is a large enough bureaucracy/army of Muslims for the Mongols to rely on, and the bulk of the cavalry in India was Muslim. So this is not likely to be conducive to a Mongol conversion to Hinduism.
In all this, I am assuming that Hindus would accept the Mongols, which may not be possible even if the Mongols accept Hindus. Hindus were incredibly hostile to conversions in this period.
