Hmm. You have, I am afraid, been a bit of a juggins! :) How on earth did you think places were sorted out - you have missed Chelsea state schools too I think. Where does your DD1 go to nursery school - do they not help with this at all?
Re Cameron House, my boys are there. The school has 20 reception places, of which at least 10 generally go to siblings. This year they had over 250 applicants, although due to people moving etc, I think they had fewer than 200 in for assessment. This is sadly moderately typical for small sW3 schools.
Having said which children do of course end up with multiple places, so there is some clearing, although these places will at first go to those who have been to assessment.
Don't rely on plans to use reception as an entry route - lists for Thomas's, GH etc close at about 1 year old - you might get lucky but this is unlikely to be a golden bullet. Again, places clear - I messed up for my first, putting him down for schools when tiny but leaving nursery schools until about 6 months, not realising the order of priority, and he didn't get a place at any of them on first pass. I finally got a place for the September only in the June.
It is all ridiculous.
Now, the key thing is to (a) don't panic and (b) be madly persistent. When places for sept come up later in this academic year, which they always do because of people moving, schools generally only make the most cursory of efforts to phone down the old waiting list. If you happen to call just after the place comes free, you can pounce. There is a boy in my DS2's reception class who popped in this way. Get your story together (I'd lean on a new to the area angle, rather than just didn't get round to it). Don't be embarrassed to call weekly - just charm the admin lady and keep it jolly.
The bigger schools are definitely most likely to come open, and Hill House is renowned for not turning people away (not an indictment - it is just very large and has a pretty transient client base, and has a fair bit of flexibility in how many it takes in reception because of the way their classrooms are set up), so that is great you have that sorted. If you prove not to feel wild about it, you have a second bite when you do DD2 as a two-fer deal, but for goodness sake get her down for some now! As previous posters said, Newton, Thomas's (Battersea or Clapham, or even Fulham) are worth a go. Spread your net wide, decide quickly if you are interested in the school, and just call round until you get lucky.
Finally, I have never heard of 4 year olds left roaming the streets for lack of a school place, so it will work out, although it is no fun to deal with at the time. Loads of luck