dreaming, I don't know about all of SW London! Only our school, and the other two local primaries. But I do know most of the KS1 parents at our school, and those who could afford tutors have discussed it with me. And I have friends at the other two schools, and they feel the same.
That said, I did hear from a friend with dc at a Richmond school that half her ds's class was being tutored - but that was for the 11+, not younger.
In my daughter's Y4 class, there is one child being tutored for private school entrance - as he is by far the youngest, and needs help in one area - and one who had confidence issues with maths, and is tutored for that. But will be going to state school, so no test issues at all. My own dd had a term's tutoring at the start of last year as she was bursting into tears over maths homework, but now she's full of confidence - no longer. She will, however, probably have a year's tutoring for 11+ so she can get her head around stuff her primary wouldn't prepare her for.
There is one child being tutored in KS1, and that is because he is the only child in the school applying to go to one of the top local preps - several others are applying to Latymer at 7+, and they are not being tutored. Nor were their older siblings (who got in).
And loads of dd2's friends from state school nursery went off to PHS and various preps. No tutoring for 4+, though (aptly, by the sound of other posts!) lots of practising cutting out with scissors 