I can't say that I care for the outbreak of 'Mumbai' that has been pouring from the telly since those terrible bombs in Bombay. Why should we suddenly call it Mumbai any more than we should now call Burma Myanmar? Twenty years ago there was a passing vogue for calling Cambodia Kampuchea.
The dictionary that I referred to said that Mumbai should be pronounced 'Moom-buy', and that is usually what folk say. The funny thing is when broadcasters want to be more ethnic than the Indians and say 'Poon-jab', when every Punjabi pronounces the first vowel as in punch (or a little more towards the vowel 'a', but in the same phonemic slot). Indeed the words are famously related, the Punjab being the land of the five rivers and punch apparently having five ingredients. I couldn't name the rivers or the ingredients.
The language of the Punjab (or Punjab if you want to make it sound more modern by dropping the article, as with Lebanon or Sudan) is a cousin of English and any other Indo-European tongue.
As for Bombay, the name seems to be Portuguese, bom bahia, 'good bay'. There is indeed a lovely bay or two there. I think Catherine of Braganza, Charles II's wife, had something to do with Bombay coming under English control. The words bom-bahia would not be transparent to most English-speakers. Mumbai seems to come from the goddess Mumba, quasi Maha-Amba, 'the great Amba', a renowned Hindu deity.
Opinions differ about whether people called Bombay Mumbai in the past. In this case onomastics are unusually fraught with chauvinism. A typical comment on Mumbai that I stumbled across goes: 'The renaming rests entirely on the goon power of the Marathi colonists' Brown Shirt party, the Shiv Shena.' Dear, dear.
There has been quite a musical chairs with Indian place names. Madras is officially Chennai. Calcutta is Kolkata, which is like the Welsh calling a taxi tacsi. Calicut is Kozhikode.
There is no simple way of decolonialising place names. One problem is rivalry between different ethnic groups about what the new name should be. Tamil nationalists were keen to call Madras Chennai, but some say that Chennai is not Tamil in origin and that Madras is (being cognate with the
Tamil for 'honey'). I don't know.
But we can't be expected to examine the rival historical and etymological claims of foreign place names before making an autonomous choice of which version to use. Every language has its own words for important foreign place names, and it is unfair to turn them into political shibboleths.