Hi OP. I am originally from the US but have lived here for over 20 years. I think many of the areas suggested to you here would be no brainers if it were just you and DH coming over on your own. The school situation in the main London boroughs is no joke.
I think you will very much have to decide what your priorities are - Is it the area and then hope that eventually schools work out if they don't immediately, with you prepared to have two children in different schools if necessary and move them during your time here (since you've missed the allocation deadline)? Or is it the schools and you are willing to compromise on the area? Your budget is very healthy if it's for the flat alone but not if you are having to consider sending one or more children private if you can't get a school place immediately.
I would decide on a few areas and then call the local councils to see which schools have spaces for children your age. The good news is that every state school gets assessed by Ofsted and you can find the reviews of the schools here:
www.gov.uk/government/organisations/ofsted. Just look under "Find an inspection report."
My experience of private schools at least in our area (not Central London) is that many of them require you to put your child's name down virtually from birth and may, even for children your DCs age, have some sort of assessment procedure. There is a good chance the ones in the areas mentioned here are also oversubscribed, but maybe I am overstating this as because of the high number of expats people do go in and out. It may be a question of waiting a term though. You can get a log in for the Good Schools Guide online and that will give you a feel for what schools are where and what they're like.
I have known two people with children in ACS Cobham in Surrey - they were really happy with it, but their companies were funding it. My colleague who looked at both ACS and St John's Wood said that SJW is much harder to get into.
For what it's worth, I like the Barnes suggestion best. I have friends there who are very happy with the local schools, lovely community, mix of expat groups.
I really think becuase you have children your DH's company needs to be helping out with a relocation consultant. If it were just the two of you, or you didn't have to think about schools for a while, you could basically just get off the plane and wander across to any of the places mentioned here and have a lovely stay.
For what it's worth I wouldn't bring my children up anywhere else but here... best decision I've ever made.