The mention of an area being "Multi-cultural", as pearlym did, is usually implying that this is somehow a virtue - such vibrant areas being so much more enlightened than the dull all-white primaries that still exist. Otherwise, why mention it at all?
I'm sure the OP is as in on the whole liberal double-think as every other hypocrite, but I'd love to see an honest answer to a genuinely naive Kiwi. Could go something like this:
Be aware! London is no longer an "English" city. White children with English as their native tongue are a minority in London. That's Greater London, which is not some core inner-city but a huge county, with twice the population of New Zealand, which includes vast swathes of surburbia and even farmland within its boundaries.
To a large extent, what might be called the respectable working class of ordinary folk has moved out, except in some unfashionable outer suburban districts. They have moved either to towns in the surrounding "Home Counties", or out of the South-East of England altogether.
Some of the professional middle-classes remain in London, in enclaves where it is prohibitively expensive to live. They either educate their children privately, or game the state schools admission system to get their children into "good" (read white and/or middle class) schools. This gaming gets even more intense by secondary education, as adolescence kicks in, and ones DCs might suffer worse than merely having their learning held back. A lot give up at this point and move to Bucks or Essex or somewhere "for the grammar schools".
That said, if you are moving halfway around the world you should have a decent enough salary to live somewhere not too unpleasant. And behind the latter day Tower of Babel reality, London i still a great city. But so is Rio de Janeiro or New York, and London should be approached with similar caveats to those places.
Can't advise you on schools, as I live in a provincial backwater, and thus send my children to the (undersubscribed) primary nearest my house, thus avoiding the endless angst of the Metropolitan liberal. But MN is the right place to find help, and you'll soon pick up the codes and hidden subtexts in each thread.
Good luck!