Mumbai village was always named after a Goddess, even before the Brits decided to build a trading post there.
The Brits were actually trying to pronounce Mumbai, could not accomplish the feat, and instead named the place with the closest pronunciation they could come up with, as Bombay.
The natives of Bombay, even during British occupation, would colloquially refer to the city as Mumbai. My grandparents used to live in Mumbai for a while and not once did they refer to the place as Bombay at home.
Same goes for Bangalore(it was always Bengalooru in local lingo), Calcutta(Kolkata), Trichinopoly aka Trichy(Thiruchirapalli) etc.