Ahahahahahahahaha!
Your views of the wider world are so narrow you cannot see out of the tunnel you are in, Xenia, Londonlottie and the rest of the SE posse.
Perhaps we need to add to advice that the best thing a mother can do is work full time earning over £100k a year in terms of best benefit to her children but also move to the SE where the chidlren will get jobs, higher aspirations and the best schools.
My parents are working class, I was raised and educated in the North, our school was crap - my siblings and I should have stood no chance by your standards. So how do you explain that I managed to obtain a degree in a hideously difficult subject and went on to become an airline pilot and now live wedged between a retired Blue Chip company multimillionaire and a TV presenter known for a program with the word millionaire in the title. (Obviously when I say 'wedged between' what I actually mean is my land butts up to their land. We wouldn't want to actually have to see each other!) My brother has made his way in the world starting on a market stall in a Northern town market with few qualifications and now lives abroad in one of the most exclusive addresses in the world - his 6 figure salary allows him that luxury. My sister has opted for freedom over income and in running her own company she can both earn enough to live well on and spend as much time as she wants with her children and pursuing her hobby, which she is a national grade judge in as well as a successful competitor. I have competed in my chosen sport at national level, and my brother is an advanced instructor in his sport. None of us compete in a sport that you would find being offered at a state or a private school. My brother's predilection for the foreign laydeez means he is now fluent in Spanish and Russian.
I like to think we are a fairly unusual family. The aspirations and ambitions of my parents, my housewife mother in particular, meant we worked for our lot, and found that sheer bloody hard work and determination can fill a gap left by our lack of SE private schooling. And I know that although we are fairly unusual, there ARE other people out there who didn't go a fancy school and who are Northern (and you can't even tell with some of them! Not a flat cap or whippet in sight,) who have achieved more than your 'I'll send them to an expensive school - that should be OK, right?' gang. Frankly these families' children ought to have better school grades, because the parents have no life now they have moved to the SE, mortgaged their souls and have nothing else in their lives than working, writing cheques to a school and spending time with their children.
So, what is the end ambition that you are aiming for here? Say you work your fingers to the bone, buy that north London house, send your daughter to Haberdashers' Askes, and what will that actually gain her - a comfortable life full of enjoyment and a sense of achievement? or a desperate worry that she will also have to sell her soul to gain the same advantage for her children? WTF is the point of that?
There are more things in this world than money. There are more things than the South bloody East of England (a hellhole, in my opinion.) There are people out there who develop themselves through travel, adventure, hard work, ambition and many, many other inventive ways. But you are so stuck running on your that you are unaware of this. I pity you your lack of imagination, but generally your loss is my gain so hey, ho.
Take a look - you don't need a private education and an Oxbridge degree to fly high. It would appear - shock horror - that you don't need any kind of degree at all, and even a B grade is good enough.