Hi, do people look at the name of your referees (i.e. if he/she is a senator, or someone who is a Harvard alumnus, etc.) more or at how well the referee knows you (even if the referee has no LLM herself? Or what if she has an LLM but she was not directly your boss and you were just colleagues at work?) Do UK universities (UCL, Essex, for example) always ask for an academic referee?

Also, I want to apply for scholarship. Should I use different referees for the application for admission and for a scholarship?

A referee's fame or lack thereof shouldn't have any bearing on how the school will look at. Nor will having an LLM or not.

Instead, the qualities they'll be looking at are (1) their relationship to you - did they supervise you? How recently? (2) how well they can speak to your capacity to pursue an LLM, and (3) what exactly they say about you. If you're considering asking a colleague, make sure that's acceptable with the schools you're applying to - often they'll need your recommenders to be direct supervisors.

In terms of scholarship applications, each school will have its own requirements. Some won't need letters of recommendations to apply for scholarships. It depends on which schools you're applying to.