Either way, considering that your child is going to be starting this coming September, you really need to look hard at what actually has spaces available. In the state sector you will only be able to apply when you have an address to apply from and particularly in London, you might find that limits your choices severely.
This might well apply in the Private sector, although you should be able to apply for a place from overseas so that will help.
You have to take a lot of factors into account and consider the long term. If you start in the private sector, your child could find it hard to adapt if you then have to pull them out. If you have the funds to cover the whole school career, then I would suggest that in your case private may be better.
State schools are well used to taking non-english speakers, private schools may well expect a certain level of ability before starting.