One thing to bear in mind - per square foot London property is significantly one expensive than Manhattan. What are your size requirements? Will you be happy with a 700sqft 2 bed flat or would you prefer something larger? And is your expectation to live in a very middle class area or are you happy somewhere more mixed? Canary Wharf (the immediate area) is populated by business types with lots of internationals. It's very clean and sterile, lots of nice restaurants and wine bars, nice shopping mall. Feels very corporate, just as you would expect. Expensive flats in skyscrapers. Greenwich is very diverse - rich bits, poor bits, street markets, tourists, lots of period properties which are much cheaper per square foot, parks, museums, the river, more bohemian...... Very, very different areas.... Also very close to each other by public transport.
With regard to schools, it very very much depends what you are looking for! There will be international schools teaching the US curriculum if you would like that, but if you want a UK curriculum you'll find everything from good state schools, alternative schools, religious schools, independent schools, independent prep schools..... And nurseries which will suit each. I would say though, that if you want a Prep school then you will stand the best chance of getting in if you register now for their nursery departments with about 6-8 schools. It seems ridiculous, but many kids will have been registered for the school's Reception Year since birth and some schools will have entrance tests for Reception even if the child has attended their own nursery. Also, there may be a few that you decide you don't like once you have actually visited them!
If you want a State school, you need to read the admission criteria carefully. Many will operate on a home-to-school distance basis, meaning that unless you have (for example) a child with special needs for whom you have completed the necessary paperwork then you will need to live within a certain distance of the school to get a place. AFAIK in that area of London the distance will vary from roughly a mile right down to 200 yards, depending on how popular the school is and how many places they have. This means that if you are fairly picky about the school it may be wise to choose the school first and then find a house within the necessary distance..... Sometimes people are very picky about state schools because the quality varies so massively - one area can have some outstanding ones and some truly terrible ones.
If you want a regular private school or a religious / other specialist one then it may be easier.
Also, would you consider living outside of London and commuting in? It's a million times easier with schools and you'd be able to get a much nicer property! Feel free to message me if you want more info.