It is kind of funny that only Londoners or Britains claim that London is the most diverse city in the world (which is a bit exaggerated because 70% is White European). Of course it is very multicultural, but not as much as NYC etc. Supposedly, there are over 300 languages spoken in London, in comparison, NYC has 170.

They surely include every minor Indian dialect in this London stat.[/quote]
Hey, I think I can help you on this one. I am a Londoner and I have just come back from a 4 month work-related stay in NYC. Please note that I am a foreigner in London myself and that I actually prefer NYC to London.
However, when it comes to diversity I can only confirm that NYC does not even come close to what London represents.
It is true that NYC might have a larger non-white population (therefore is less racially dominated by one group) but because of its geographical location, less stringent visa regime and membership in EU London gets people from many, many more areas.
In NYC it is pretty much about Mexicans, Puerto Ricans (who are actually US Citizens and who seem to have stopped coming some 20 years ago), Koreans and the Chinese, ppl from Russia / Ukraine and Haitians / Jamaicans. And then you have the proper Americans who admittedly are also a varied bunch but in the end speak the same language and share much more than, e.g., people from neighbouring European countries.
London gets people from everywhere and in much more even proportions which you can really feel while in the train / at your work place or in the street.
Moreover the immigration in London is so much more recent and fluid. So many people arrived in the last 10 years due to the (now past) economic boom and so many people come for a short period of time, for a year or two, just to be replaced by others.
The only ethnic / cultural group that is a bit missing are Latinos.
The bulk of them here are Brazilians. As for the rest I provide you a bit of overview below. But I stand by the fact that London is way, way more diverse than NYC. In London immigration is happening /exploding as we speak, in NYC it was 40-20 years ago....
Below is a small outline of where ppl in London mostly come from-
Africa (West mainly Nigeria, Ghana although a lot of people from French-speaking countries,
East - Somalia, Eritrea, Ethiopia,
South - Angola, RSA and Zimbabwe, mainly Whites)
the Caribbean (Jamaicans followed by the rest, similar to NYC)
Middle East (Kurds, all sort of Arabs and Berbers, all ethnicities of Iran and Afghanistan)
Eastern Europe (led by Poles, then a plethora of people from the remaining 12 new states and even more ethnic groups)
Western Europe (French, Italians from the South, Irish, Spaniards, Portuguese and professionals or people coming to study the language or hangout for a year just for adventure from other EU countries)
Indian Subcontinent (we are not talking about dialects but languages in their own right with their own scripts and independent histories longer than the one of English like Hindi-300m global speakers,
Punjabi-70m, Gujarati-50, Urdu-over 100m, Tamil-70?, Telugu-70?, Bangla-over 200m?. Please
note that these people look phisically different, I could tell one group from the other by their faces.
Australians and Kiwis (huge communities, not too many Pacific Islanders)
Asia - Hong Kong Chinese, Philipinnos, Language / University students from Japan, Korea or China / Taiwan