In any UK LLM, not that much. I have personally seen students from all kinds of unis. Do well at university, that's far more important. Just a few years ago, the first blind person from India to get the Rhodes Scholarship went to Oxford for BCL and MPhil. He attended the University of Nagpur and had a really killer profile.

At the same time, though from my own experience of having taken a UK UG degree and then having done an LLM and having seen other students experience this as well (and forgive me if this sounds obvious), some at least of the unis have implicit biases in preference for their own LLB students to do an LLM compared to others. For example, with the much-vaunted Oxford BCL, I once saw a student with an Oxford undergraduate BA in Jurisprudence (which is what Oxford calls its UG degrees), who got 2:1 in Oxford where the condition was to get a first, still getting into the BCL - the offer wasn't rescinded (though because of Oxford's system of making students take most of their degree module exams at the end of third year in a three-year degree, I'd say that anyone with a 2:1 Oxon BA Jurisprudence would be more likely than not in any other uni to get a first - they'd sure as hell have all the intellectual ingredients to get it). I don't think anyone, even from NLS, would get wiggleroom like that (though, from what I have heard of the lifting of academic standards to sky-high levels under that former DPhil supervisee of Paul Craig's who is now their VC, that is a bit of a shame, and they probably should get that similar kind of license to an Oxford student). Similarly, the London universities offer scholarships to students who have been to their own unis or other unis within the London university network which others are not eligible for (though these scholarships are more in name only, and literal peanuts in the larger scheme of things - a couple of thousand GBP max). For example, a guy I know that went to Queen Mary for LLB and got a first got a £3000 scholarship for UCL LLM, but would have gotten £6000, had he chosen to stay on at Queen Mary for LLM.

Oxford - In my opinion it's one of the most important factors. See their profile books for the BCL and look at the Indian Candidates. Literally all of them are rank holders from NLUs or JGLS. Maybe once in couple of years someone from SLSP or GLC gets in.

[Folks from lesser known colleges don't lose hope though just means that it's statistically less likely; not impossible. There is a first to everything. ]

Cambridge - Does not matter that much. They are more concerned with your application holistically and are a less selective that the Oxon BCL.

NUS - is not that big of a deal to get into. . It's called the Jindal for LLMs. If you have the money and a decent profile, you'll get in. (don't get me wrong - it's a great institution - just much less selective as compared to Oxbridge)

to answer the second limb of your question "if a student maintains a strong academic record (top rank), secures quality internships, builds research exposure, and obtains strong faculty recommendations, how much weight does the college “tag” actually carry in the admissions process?" - not very much, a candidate from a mediocre institute but with a killer profile (think - rank 1, excellent internships, publications, moot wins, RA ships and a authentic SoP and LORs that speak about their abilities uniquely) would beat a mediocre student from NLS 9/10 times who has a average rank and did not do anything much during law school.

Differing opinions are welcome

Does the institution where one completes an undergraduate law degree in India significantly impact admission chances for top LLM programs abroad (e.g., Oxford, Cambridge, NUS)?

If a student maintains a strong academic record (top rank), secures quality internships, builds research exposure, and obtains strong faculty recommendations, how much weight does the college “tag” actually carry in the admissions process?