To study English Literature in India is as important as studying English Literature in any other country. In my opinion, the grand prize of English Literature can be summed up in one word: Shakespeare. Shakespeare alone (even if you consider him to actually be Kyd or Marlowe) makes English Literature worthwhile studying. This is not because England is some kind of extraordinary country. No. It is simply because the literature of Shakespeare (including both his plays AND his sonnets) is so incredibly profound and so astonishingly universal that all cultures have been able to identify with the many characters enmeshed within it. In fact, even if you consider "English Literature" to be ANY literature written in the English language, . . . the word Shakespeare can STILL sum it up nicely.
Therefore, there isn't one specific "skill" that someone learns by reading only English Literature. (Unless you want to point to something specifically English, such as the type of sonnet called the Shakespearean Sonnet.) It is less a set of "skills" as it is a set of amazing authors: Bede, Chaucer, Shakespeare, Keats, Shelley, Austen, and Shaw just to scratch the surface. In fact, I would argue that one would be quite handicapped indeed if English Literature were the ONLY kind of literature taught in high schools and universities.
Consider the course of study for most high school students in the United States of America: 9th grade, grammar/writing; 10th grade, American Literature; 11th grade, British Literature (English Literature); and 12th grade, World Literature. To be well-read, therefore, an American student should have knowledge of Indian Literature as well! And I would hope that the high school equivalent in India would include a full year of Indian Literature and that perhaps English and American Literature would be grouped into one, or perhaps even included in a World Literature class.
I believe any of us would be quite narrow-minded if we knew literature only from our own country; therefore, kudos to those who stress the importance of World Literature to expand student knowledge!
There seems to be two different questions present here. The first would be to study English Literature in India. I would think that this could be a very worthwhile pursuit in a couple of domains. Initially, the richness of ideas as present in English Literature, like other branches of Literature, is powerfully compelling to study and analyze. At the same time, to study English Literature in a nation that had been controlled by England for centuries might be intellectually profound. Does the view of literature from an imperialistic nation impact the reading of it? This could be quite an interesting topic to examine and study. The second part of the question brought out in the second part is the basis of all post colonial literature. The expression of cultural narratives in languages or ideas indigenous to the region helps to bring out the idea that literature under colonialism could be rich and powerful in its own right.