I was also going to say that academic, competitive schools are going to be hard to get into - especially as they do entrance exams. It's a jungle in London, especially SW.
I do not for the life of me see schools like Dulwich and Kings accepting late comers, unless your DC is spectacularly gifted. I know DC who leave perfectly wonderful schools like Trinity/Whitgift to join these two schools. Parents will leave a sure offer of another school, to join them if they were on the waiting list. You may be lucky, but manage your expectations. What I'm saying above will apply for Westminster, St Pauls, City etc.
London no longer has 'back ups'. So schools like Emanuel and preps like Newton, Wetherby etc. are getting quite competitive to get into.
Even for your Y3 DC, there is a high competition for those who sit exams at 7+, 8+, 11+ and 13+. Your best bet, is to take your DC to a prep that is more naturing without entrance exams, but is a great feeder (say like Hill House, Eaton Prep etc). I would then look into getting both DC at a prep for now. You have to realise that the 13+ schools wrote their ISEBS in Oct/Nov of Y6 (annoyingly, it's at the same time as the 11+ entry). But there'll be late intakes from year 7 sometimes. So your prep Head will likely be well connected and be able to get your DC into this second stage.
You have a lot of work ahead, but ring around. It's possible to get both your DC into one prep with the view of moving them on at 13+. The Clapham/Wandworth or Central London area will have a lot of preps you can try who may eventually feed your DC to your ideal schools.
Someone posted you a great link on this thread, I would look into your nearest schools and research those that catch your eye and ring them.
Good luck