What were the British known locally as in India?
It is unclear what precisely the question is asking, but the following discussion of British rule in India will hopefully provide the necessary information.
British colonization of India evolved gradually, beginning with the establishment of trade links in the early 17th Century. The British established the East India Company to facilitate and manage trade operations througout Asia, and it was this "company" that set up operations along India's coastal areas.
During the course of the 18th Century, British political control over India expanded and India was more formally colonized and integrated into the British Empire. The East India Company continued to play a major role in British colonial rule, and company officials became "governors" of various Indian provinces. By the middle of the 19th Century, British administration of India had formally shifted from the East India Company to the British Crown, and in 1858 what became known as the British Raj was established. The period of the British Raj would extend until India's independence in 1947, and the subsequent break-up of India into India, Pakistan, and, eventually, Bangladesh (from 1947 to 1971, Bangladesh was part of Pakistan; following a brutal war of liberation, the independent country of Bangladesh was established in 1971).
In answer to the question, then, either the East India Company or the British Raj would be the likely response. In the event the question is seeking Indian slang for their British colonial rulers, either "baba," or "father" might be the answer.