To fulfill the letter if not the spirit of the challenge:
During the East India Company period, Bombay (as they call it) is designated as a special administrative capital of EIC administration. An Anglo-Indian minority/administrative class emerges there. Bombay remains the capital when Britain assumes imperial rule. A larger Anglo-Indian class emerges, being involved in administration, mercantile activities, some highly-skilled trades etc.
By the early 1910s "Bombay" is seen as the Anglo-Indian city. They form a significant proportion of the population (20-25%), though the total population of the city is a bit smaller than OTL.
Come independence, Bombay is seen as too extensively Anglo-Indian to hand over. Britain retains it as a small enclave in an otherwise independent India, calling it British India. Unlike Portugal in Goa, British rule and reputation is sufficient that no military takeover is attempted.
During and immediately after independence, many Anglo-Indians from across the subcontinent move to "British India", while many Indians move out to avoid ongoing British rule. This leaves a slight majority of the city's inhabitants as Anglo-Indian.
India then renames itself Bharat to avoid any association with Britain. In popular parlance, British India it is simply called India, and eventually that becomes its official name.