So, I know the Brits in India was not always a harmonious relationship, but there have been some good things to come both ways through the few hundred years since "Johnny Company" schmoozed their way in. Language has just been a small part of this, with the language of Indian government being English as it unified all of the regions who speak many languages and thousands of dialects. However, this was not a one-way process; there are lots of words appearing in English from the sub-continent.
Lets start with doolally, or doolally tap to give it the proper name. Deolali in India was a transport camp, and the word tap comes from Sanskrit tapa meaning fever. It meant madness through being in the heat of Deolali waiting to be sent to station somewhere else, so someone goes a bit doolally, they are a bit 'fevered'.
Other words, some you might know, some you might not are in daily use.
One might inclined to slip on pyjamas, which the lightweight trousers worn by men (one still buys a pajama suit, in fact I have a couple..). These are worn by men and women all over the region. Then you could leave your bungalow (originally from Bengal, a style of house there), and sit on the verandah. While there you could watch Avatar (avtar is Hindi for a form of God).
Even a catamaran comes from Tamil (originally kattumaram meaning tied wood), and of course settling down for a curry (kari / karhai) with a mango chutney (mangaai, chatni, meaning crushed) is a nod to Tamil.
Star Trek has also used Hindi / Urdu words, I suppose they sound sufficiently unusual: Ferenghi actually means foreigner, and Jemadar was a rank in the Indian army.
These are obviously very limited examples, there are actually hundreds of words, some obvious, others not so obvious (see http://en.wikipedia.org/wiki/List_of_English_words_of_Hindi_or_Urdu_origin), and of course many other languages have contributed to the language. Apparently the letter J did not even exist in English until the French invaded.
I have been delving into the world of the British-Indian relationship as I am researching some of the history from a genealogical perspective, but I find the deeper I dig, the more interested I get in all things about this. Sadly my granddad died when I was 10, and I missed the chance to ask him about his experiences, and my grandma was gradually afflicted with dementia before I became interested in this part of their lives. I am putting bits and pieces together, and my ultimate goal is another trip to India, particularly to Bellary, where my great-grandparents are both interred, probably in a 'plague pit' as they died from cholera. I hope some people will find my posts about this sort of thing as interesting as I do! More to come.