Instead of going west, Tamerlane decides he is going to destroy the Hindus of India and convert them to Islam.
He more or less succeeds and then dies from some disease in India.
How does this change world history?
Commissar
If it happened, hugely. However two questions.
a) At which point do you have Tamerlane do this? He headed west against various foes several times. Suspect your referring to towards the end of his career, when he launched another series of attacks on Georgia and Armenia then defeated the Ottomans. However he went west, further north, other times in his attacks on the Golden Horde. It was in the latter period, possibly because he was more aware of his impending death, that he became more savagely religious.
b) Do you actually think he could achieve that? He sacked Dehli OTL but that was a single campaign and while he slaughtered a lot of people in the immediate location and region vast areas were never directly affected. Also he seemed quite skilled at picking times when neighbours were weakened or distracted by internal unrest or a change of ruler. If he goes in earlier he might find much stronger opposition from the Dehli Sultanate. Also if he's openly following the path of forced conversion or slaughter then he will face a lot more opposition from both the Hindus and other non-Muslims and probably a fair number of moderate Muslims as well.
Furthermore, even if he manages to reach and attack southern India, beyond the Indus and Ganges plains, that will be a multi-year operation, probably a decade at least. [Think of how, despite attacks over a couple of decades, the Georgians and Armenians managed to continue to exist, albeit probably greatly weakened]. If nothing else he can't afford to leave the rest of his empire for this long as it will see both external attack and internal unrest.
As such I don't think the sort of thing you suggest is possible. Possibly more attacks on northern India, causing yet more death and destruction. However I suspect those would be yearly incursion as he virtually always returned to his capital each winter, which would limit the depth into which he could penetrate India.
What is possibly the most likely case from something like this would probably be the drastic reduction in Islam in India. The southern states are going to be largely unaffected, other than being boosted by refuggess from the north while the heartlands of Islamic power in the north are likely to be gravely weakened. You could get a lot of conversions at sword-point but once his empire withdraws or collapses under the strain many would go back to their former state and Islam would be detested and loathed throughout much of the peninsula.
Steve