You could also have a look at St Paul's Cathedral School for the younger ones at least (and in fact they do go to 13 so it could be an option for your 11yo to buy some time while you figure out what is best for him academically if he doesn't get a place in your preferred senior schools right away). It will be pretty close to your work if you are at Liverpool Street, so you could all commute together. My office is very near LS and it takes me about 10 minutes to walk to the school to pick them up if I hustle. They have wrap around care 7.45am-6pm from Y1 (though I think in Reception it's only 8.30-5). It's full but I don't think it's super-hard to get into (everyone we knew 5 years ago who really wanted it got it, even if they were wait-listed first), and might give you a bit of breathing space while you research other options (and maybe as a back-up for your 11yo if you are trying for other schools as well).
Our two DC (one girl one boy) have both gone there for the last 4 years and we've been really happy with it. DD just left for senior school and DS is still there. It's got a lovely atmosphere, the kids get to use the Cathedral a lot and feel a real connection to it (even the non-choristers), and it really lets the kids play to their individuality - supporting those who need an extra stretch as well as those who need an extra hand.
Historically I think they have possibly been a bit casual about the "Prep" part of their job (i.e. great on science, music obviously, art, history, geography, drama, etc. but not so hot on 11+ preparations) but the new Head has been making a few changes that are improving that without turning it into a hot house. Having said that the kids in DD's year have gone on to some quite academic senior schools (including Highgate, South Hampstead, City, Westminster) - but I think probably all of their parents have supplemented school with some tutoring.
Separately: if you are thinking about private schools further out (e.g. Belmont/Mill Hill, Habs, NLCS, St Margaret's Bushey, Aldenham, Haileybury, or whatever) as options, it's worth looking at their bus routes before you commit to somewhere to live, as even quite a small shift in location can make a quite chunky difference to commuting times for them.