Is it not a good idea to visit London at the end of July? I know that's when the schools' 6 week summer break is.
I think it is worth placing a call to an education consultant now. I think that most schools are staffed for the first week after term ends and then a week or so before school starts but in between it will be caretakers and painters in my experience.
It is really worth visiting schools when they are in session as you get a much better idea of atmosphere and how the kids interact with staff. I would recommend that you come over for a few days in early June after the summer half term and visit some schools on your own or with the 2yo and get a feel for areas/schools. If by "next year" you mean January 2017 rather than school year, then it might be worth booking a weeks holiday en famille this summer and getting a feel for where in the city appeals most.
Private schools average £4-5k per term so that's £15k pa, per child. You may get a small discount when your 3rd child starts but it will be imperceptible.
Therefore in my view you need to make a call sooner rather than later on private v state schooling. With your housing budget [assuming it can be adjusted in favour of schooling and isn't set by your husband's employer] you have plenty of choice.
OP - with children of that age and your budget I would personally think that having some reasonable garden space that comes with the house is essential for your own sanity. Trying to keep an eye on three small children simultaneously on your own in a public London park is hard work! So while the Regents Park pad is stunning, I'd run a mile from it personally for that reason explicitly. Plus I have lived in the area when single, central London is fabulous but air con is very rare. When it reaches 30 degrees central London is airless, polluted and you will want to be anywhere else. So choose your property carefully.
You might not wish to drive initially, but access to a car/parking to get out of London is wonderful. We have a car but have done less than 10,000 miles in 3.5 years as an example. We only use it to leave London ! Public transport is excellent, usually much faster than driving, but if you want a weekend away or half term by the coast, you will end up packing half the house into a car to cater for the unreliability of the British weather. Plus it is really incredibly easy to drive to France which becomes very tempting :)
It is possible to do the vast majority of your grocery shopping easily and painlessly on-line in the UK so on the upside, you won't need a car to schlep to the supermarket twice a week.
I know the area your husband will be working in very well. While SW London is really lovely Chiswick, Battersea, Richmond, Fulham are all more than a 30 mins commute. The first three in particular will require multiple train changes so it's not "just an extra 10 mins" - you can't board a train and just read a book. Chelsea/Kensington is borderline. Fine when the Circle/District are running but there are frequent issues and then it feels like a million miles away. Doable though particularly if your DH has to be at work v early before the trains get crowded and failures start.
Easiest access to that part of London to work is:
Liverpool St station [Central Line [red]
Circle [yellow], District [green] and Metropolitan [purplish red]
Aldgate St station is end of the line for the Metropolitan so your DH can simply hop on a train, get a seat and relax until he changes at Baker St to head north towards Maida Vale/St John's wood on the Jubilee line [grey] . These areas are close to Paddington station which is where the Heathrow express train terminates.
London Bridge and Cannon St stations are a ten mins walk so that opens up a lot of South East London. If your husband's employer will sign off on using City Airport to NY route [pricier than Heathrow] then I would really look at Blackheath which is v chi chi and has several private prep schools and a lot of open space. Even if they won't every 4-6 weeks is bearable, more frequently then head west.
Intersecting all this is the City branch of the Northern line [black] aka "The Drain". Not a pleasant commute by all accounts but intersects with London Bridge/Bank/Moorgate which are all stations close your DH's office. It opens up Islington/Clapham and Balham as family friendly chic suburbs within central London.
Re Church I'm not familiar with your branch of worship but going on Presbyterianism alone these seem like options. It depends on whether you attend frequently as a family as to whether your church would heavily influence your choice of residential location.
www.londonfreechurch.org.uk/services.php
this is on the central line in the financial district but further west that Aldgate. It's easily accessed from most locations but is not in a residential area.
www.crowncourtchurch.org.uk/where-to-find-us/
This is in Covent Garden so heart of the West End. Accessible but perhaps a bit jarring to leave a church service and walk straight into the midst of Mammon.
stcolumbas.org.uk/?page_id=460
This church is in Knightsbridge so is surrounded by the parts of London that your housing budget would support 
Final word from me which is completely off topic but central imo to your quality of life in the UK as a family.
Check that your DH will get British public holidays and a European holiday allowance if he transfers here. 5.6 weeks/28 days is the norm here incl public holidays [20+8] and people take them. City employers will usually offer 26-30 days plus public holidays.
American employers tend to either offer the full UK entitlement plus USA holidays if the office is closed OR worst case scenario, 10 days holidays but NO USA public holiday and USA employees are often expected to work UK public holidays if the USA markets are open. The latter is really rather rubbish and I know someone in that position personally.
European holiday entitlement is what most American miss the most when they move home and while your kids are so young it would be lovely to have it particularly when you are living away from friends and family and Europe is on your doorstep.