Do I need B.A. LLB (hons) from to pursue LLM from UK, or will just B.A. LLB do?

Usually just a first degree in law will do, but it depends on the school.

For instance LSE looks for 'good grades,' which they qualify meaning 'first or very high upper second' in terms of the UK grading system.

Sussex asks for an upper second-class (2.1) honors degree or above. Bangor wants a lower second class degree.

Not sure how your grades stack up, it depends on your grading system. You can usually ask the schools you are interested in, directly, and ask them if they would realistically consider you as a candidate.