I think it's impossible to find a central London indie that is similar in student population diversity as London population.
Biggest barrier to diversity is > £20K (and rising every year) annual fee while London median wage remains around £33K (before tax). Most ethnic minority groups in UK on average earn less than the median.
Even the most financially well off, largest indies, spend no more than 10% of their annual revenue on bursaries (CLSG being the exception that I know of).
So it's inevitable that the student population will be one particular socio economic bubble. They live in similar houses, own similar gadgets, have similar vacations, wear similar clothes, have similar birthday parties and so on (with small variations, of course).
Like the OP, two years ago, we set out to find such a school and now accept that such a thing does not exist.
There are of course small variations within schools -- anecdotally, G&L or SHHS seemed much less diverse to me than CLSG, but the differences between the schools is insignificant compared to the differences between school population and the world outside.
So we just accept now that by sending DD to a selective indie, we have inadvertently created a bubble like social environment for her. Whether this changes her for the worse depends to some extent on her and on us.