I also think that it is one of the saddest things for all of the UK that everything is so London-centric, and if there's ANYTHING positive that we can take out of the pandemic, it's that we can spread everything out over the UK! To the many other smaller but utterly beautiful and magnificent cities!
OP, we lived the dream - a house in Hampstead, a kid in private pre-prep. The reality was the grind was intense, Hampstead was not very friendly as everyone too busy (although our street was absolutely delightful), the many baby groups etc no one really chatted to each other. I felt very, very, very uncomfortable with the enormity of the gap between the rich and the poor, and how this was reflected in education. The reality, as I was discussing with a friend yesterday who is still there, is that you need 20k+ per child, plus several grand bus fees plus a nanny just for education needs. So one of you needs to be earning 120k+ just for that. Then the other one has to earn a fortune to meet the large mortgage cost, the train passes for commuting (several grand). The metro-boulot-dodo is fucking relentless.
Also, your DH must bear in mind some of the stories that come out of the admittedly fantastic private schools and SE grammars that send 20% to Oxbridge. They aggressively self-select - good luck with watching your child go through selection procedures of the guildford equiv of Highgate at 2.5 years old, as very serious teachers grade their ability to stack several bricks on top of each other and then reject them anyway (400 applicants for 9 non-sibling places in our year). Good luck with the pressure for sitting the 7 plus at several different schools, good luck with hoping that you can get siblings into the same school or that your kids aren't managed out for not hitting the high notes at 12. Oh, and helping your teenager understand that not getting 11A* or whatever the equiv is does not mean they are a thick muppet who should immediately start self-harming, and getting them to understand that there is life outside the SE bubble.
There will be tons of parents saying 'oh, but it's wonderful, once you're in', but read the 7+, 11+ and 13+ threads in the education boards, that start off all chirpy and ambitious and end up thirty pages later with a mum having to tell her bright, sensitive, highly tutored 10 year old that despite having sat exams at five different schools, despite knowing how important they are and despite studying for it for years, they have not been given a place. The effect on that on your kid!
We pulled our kid out, moved to NI, are moving into a lovely big house for less money, and our kids will go to one of the many excellent state schools with a big social mix which will equip them far better for life. They have lovely fresh air, tons of sports, great community and amazing activities at a fraction of the cost.
Obviously no one ever moves to NI without a connection, but if I were you with great jobs in Edinburgh, a beautiful international city of a manageable size, with an education system that is incredibly well thought of, I would definitely NOT move. Half of mumsnet will be envying you right now! It's impossible to describe the daily commuting grind of London (which I LOVE as a city and where I lived for 17 years - at the end even my bus from Hampstead to the river was taking 90 minutes each way! Trains were always down!).
Also, doing several highers is great as it doesn't narrow down your opportunities! I had to get math tutoring sixteen years out of school to function at business school!