I'm in SW London as well and I think you do need to think about it early in some circumstances with regard to the private schools, although I don't think you need to at that age with the state schools.
I think if you are looking at one of the schools that does selective entry, you probably don't need to worry so much: they will look at all applicants at 3.5 or whatever.
Otherwise, you will find that some schools do close their lists relatively early, particularly, as happened to us, it is a big sibling intake year. However there are 2 important factors - firstly, a lot of people are registered at 2 or 3 places and so cannot take up all places; and secondly a lot of people move out of London when their children are young. So places open up all the time.
I think most schools don't definitievely tell you whether you have a place until later on because of this. A couple of the schools we were registered for also asked us to re-confirm our interest a couple of times, which again I suspect is them managing their lists.
There also seems to be a great window of opportunity in the 4 weeks or so before school starts, when despite initially gargantuan waiting lists some places become availabile, if you have the nerve to go down this route!