The American School is in St. John's Woods and your budget and bedroom/bathroom requirements are healthy enough for that and surrounding areas so relax.
. Regarding the tube, where is your husband going to be working? You should get a tube map and a map of London so you can get your bearings. Does your husband want a commute on 1tube line or is he ok with a transfer? Also, at the moment, the tube isn't air conditioned.
Here are some property websites:
Findaproperty
Primelocation
Rightmove
This website can give you some neighborhood info:
Upmystreet
St. John's Woods is lovely and as you would expect because of the school, there are a lot of Americans there. There is a nice High Street with restaurants and shops. There is even a small grocery store off the High Street which carries some American foods that you might stRt missing like Kraft Mac and Cheese.
Other areas you might consider are Maida Vale, Marylebone and Bayswater which are still pretty close to the American School. But, do consider your husband's commute and how close you want to be to the school. They do offer a bus service for an extra charge.