We've been offered a place in our first choice state school and I was also not sure what to do, since Ds had also had a place in a very good private (also London). We are in the same position, with two Dc we can only afford private either at primary or at secondary, not both. After speaking with other people with kids in state schools we decided to accept the state school place....
What most of them say, and it's true in my opinion, is that for young kids, family/parents are much more important and influential than friends. Meaning with this that the core values come from home! Therefore, I prefer to make sure he's in a more controlled environment (private) when there is more need to it, like teenager years. Also, in Ds foundation state I prefer for him to be exposed to a social and cultural mix, making sure he's raised with good values and understanding and learning to accept all kinds of people. I hope when the time comes to send him to private school (where the core group is less mixed) his values are well settled and he has a broad social view of who and what surrounds him.
What I also heard from different people was that it is easier to work/study/help out your child at home while they are younger, and therefore, if state school means more support at home, then primary school should be easier to support than secondary. Also, you can always get the tutor if needed be.
In the end, these where the reasons that make us decide on the state school, but I'm sure others will have also very good reasons to make things the other way around...
Also, don't know where you are in London, but in our area the grammar schools are almost inexistent!
Good luck!