I'll confess that I've only read the first few posts of this already very long thread, but a few things spring to mind:
a) if your children can go to a reasaonble state primary school, you can tutor for secondary entrance exams which is much cheaper and you can do this throughout their primary school years, rather than just the year before;
b) you've mentioned that your husband is relying on increasing earnings, if this is a fairly secure career progression, eg to hospital consultant with a private practice, or to partnership in a big law firm then that's fine, but otherwise I'd be cutting your cloth to what you earn now;
c) can you move to an area with selective state grammars or exceptional state secondaries (something that Labour policitians always seem to achieve in London) - we've always assumed that our kids would go private for secondary because we both did, but in fact our local state secondary is outstanding, and the issue is that we might have to put our son into a prep school because there are too few primary school places in our area - I do think that the move from prep to state secondary is not great as your kids are unlikely to have friends moving with them and may be stuck with the 'posh kid' tag.
Private education is always a challenge financially I think, unless you're really minted, but more so if you have 2 or more kids. I was an only child and benefitted from an assisted place which meant that my fees were calculated according to my (self-employed) dad's last year's income so it was mostly affordable for them but there were certainly some sacrifices. My school did have reduced fees and bursaries for multiple kids, but I think this was mostly reserved for the less well-paid middle class professions, eg clergy and teachers, so may not apply to you (and this was in the NW in the 80s, not London today).
£12k by the way would almost educate 2 kids a year privately round here up to age 11 (would be around £13k), it's slightly horrifying that you're expected to pay that for one child for pre-prep.