Because India is composed of many completely different races, languages, and thus cultures - you can't say India is "culturally influential" because there's no such thing as "Indian culture".
Not much of that (nor Scots Gaelic, nor Irish) in the Empire.
And if local culture like Morris dancing, Caber tossing, and cheese rolling made it off the island, I never heard about it; conversely, the things which rich people in the UK did enjoy (Cricket, Shakespeare, Christianity) they did export.