Harvard p-grad here. Having a NALSAR or NLS background does help but only to the extent of being better trained and having great connections with your senior/alumni who have already done their LLM from here. That meant I knew exactly how to train, what to write and how to go about my applications. A great set of faculty or alums (who in my case was also my Partner in the law firm I was working) really helps with the recommendations. I guess the one major advantage of NLS/NALSAR supremacy has been its old school network-i.e. alumni and students...what I am trying to say is it is the students that make the university and not so much as the faculty or admin. But yes some faculty of NLS/NALSAR have been exceptionally well read and respected and yes if a word of recommendation can come from them it may make a lot of difference. Having said that my batch had people from India who did their LLB from non NLUs and I will say they were equally talented. So best of luck!