London is a great city, as is Paris, Tokyo, New York but the sheer effort of working (in a median waged job) to afford to live well with children in these cities is not worth it. I have lived in the south east but for me its no contest! Edinburgh is a wonderful city, free museums, you can walk everywhere, its cosmopolitan, beaches and countryside are less than 30 minutes away, the Scottish school system is less stressful. Over subscribing at primary school level is very very rare.....I wouldn't even worry about that. There is absolutely none of the angst I hear about regarding secondary school places, if you go to the primary within the catchment you get into the local secondary. People often chose to go to the local catholic secondary school rather than non-denominational as they often have better reputations. ( that's allowed!) .
Taking 5 Scottish Highers allows young people to study at depth and breadth rather than limit it 3 A levels. Good quality education at secondary and tertiary is well established here, and its free of tuition fees.
In fact it takes so long to travel across London, in the same St Andrews - all within 90 minutes or less, and you get to see the rest of Scotland.
You also miss out on the (daily or twice weekly) time with grandparents. I know of 2 sets of grandparents who have moved here from the SE and SW to be near growing GC.
Leave and breathe! Seriously I don't envy your decision, but when I lived in the SE I was astonished at people's myopic attitude towards the rest of the country, and almost shocking ignorance of the treasure that is Scotland!!