Do not know, before my time! I was just A child when the British left. But even during the he British colonial period, there were autonomous kingdoms existing in India. My fasther was a high ranking official in the 2nd largest kingdom ruled by the Maharaja of Baroda. And IIRC the British had no control over our kingdom.
Also your theory doe not make sense, because Pakistan, Jordan, Iraq etc also had lot of British involvement and democracies have a checkered past there. I think it has a lot more than to do with the large number of educated people in India. Universities for Ãhigher education existed in India 2000 years ago.
HTTPS://www.youtube.com/watch?v=xjB_Tf7Cy3A&feature=related
-PJ