I grew up in London and am bringing my kids up here, too.
There is always something to do, see, experience.
We can walk / get public transport anyway (much healthier and more fun than 'country' lifestyles that involve driving everywhere).
The mix of different people, cultures, languages we come into contact with everyday is enormously wide, and I really feel my children benefit from this.
Also, the myth that living in London means no 'fresh air' and freedom annoys me. I spent my childhood - entire summers, in fact - in Hyde Park, St James Park, Holland Park, Hampstead Heath, or tearing up and down the South Bank, or swimming in one of the lidos.
It's one of the greatest cities on earth, and it is what you make it.