ngrams.googlelabs.com gives us the ability to measure the extent to which a word occurs in the millions of books that google has digitised. For the Bombay vs. Mumbai question, it shows:
|The phrase `Bombay' vs `Mumbai' in books|
This suggests that the people who write books are still emphasising `Bombay' instead of Mumbai.
Turning away from books to the web, google gives us two kinds of information: the extent to which either name is used in the stock of material on the net (as of today), and the extent to which either name is used in google searches.
|City||Share in search||Share in web|
In the case of Bombay, this tells us that the old name (`Bombay') makes up 19% of google search traffic and 22% of the stock of content on the web. So this evidence suggests that the new name has been accepted the most in contemporary use with Chennai, less so with Mumbai and further less with Kolkata. I wonder why this would happen.