Someone above mentioned Millburn; yes I live in Millburn township in Essex County (which also incorporates Short Hills which has its own zipcode and train station). It's a fantastic location....15 minutes/10 miles west on the Interstate Rt 78 from Newark airport (BA, Virgin and United airlines) and approx 40 to 50 minutes into Manhatten, although I have done it in 30 minutes non-rush hour in the car. It isn't far from Weehawken.
There is the enormous South Mountain Reservation of hundreds of acres which borders Millburn, parts of Maplewood and South Orange....thousands of trees, open spaces and a great place to take a dog as it also has a dedicated dog park. Also only 5 minutes by car from Millburn is the very nice Turtle Back Zoo, which is adjacent to the beautifully renovated reservoir park with a fabulous play park for children, swan boats etc. (see www.essexcountyparks.org and look up the South Mountain Reservation).
The school district is regarded as one of the best in the USA, although neighbourilng towns eg. Maplewood, South Orange, Summit, Chatham and Madison also have excellent schools too (and are all on the MidTown Direct line into NY Penn Station (underneath Madison Square Garden) and also trains into Hoboken for lower Manhattan. Your address/zipcode will determine which school district you are in, but all realtors will know exactly when you go to look at properties.
Whenever I go shopping for groceries in the wonderful Trader Joe's on the edge of Millburn and Maplewood (owned by Aldi, it's a pity they're not in the UK) I hear an English or Aussie accent, there are loads of expats around here. (Check out www.traderjoes.com)
OTOH if you are in a town which isn't on a train line, property prices are lower (commuter parking is usually restricted to township residents and some towns have long wait lists, although they may offer a jitney (minibus) service to some parts of suburban towns. You may wish to have a look at for example, West Orange. If your spouse doesn't need to go into NYC for meetings and will only need to be in Weehawken, this may be a good option for you and do bear in mind that the further you are from NYC then it tends to be cheaper for property as well.
Millburn has a number of nursery schools (kindergarten) including Montessori, a Co-Operative nursery school (parents volunteer to help out with running it one day per month) and across the street opposite Taylor Park is a non-religious preschool at St. Stephen's Episcopal Church in the town centre and I know Brit and Aussie expats living in and adjacent towns who sent their children there. (Episcopal is the US version of the Church of England)....actually the rector, Sheelagh Clarke is from Wimbledon, London and she operated the preschool before becoming the Rector (she's really nice btw).
You can google Millburn + Preschool and see the list. There are also Jewish preschools in the area....my son's GF is a nursery teacher at a non-denominational preschool in South Orange. Also in Maplewood is a daycare operated by the 'Y' (the YMCA).
As you are in rural West Sussex, I think that you may very much like the pretty 'village centre' in Maplewood (5 minutes in the car from Millburn) as it's the nearest equivalent I think to an English large village. Like Millburn It has good casual places to dine (all family friendly) and a pub!
Both towns have a cinema and the Recreation departments organize lots of sporting events eg. Ski trips for kids in the Winter and operate the open air swimming pools in the Summer....it gets very hot and humid and you need air-conditioning due to the high humidity. It's actually very warm right now, in the high 70sF, although this is unusually warm for October. Another great place is the Connection in Summit (8-10 minutes drive from Millburn) which is a family-friendly gym and indoor swimming pool....they offer dozens of programmes for children and adults for a reasonable price eg. Dance classes, yoga, gymnastics, cooking, art, swimming lessons/fun and children's birthday parties. See here:
www.theconnectiononline.org
I would also take a look at Montclair, but there isn't as much recreational space there compared to where I'm located and the schools are not as highly regarded....but I do like to go for brunch at Raymond's in Church St. lol!
Happy to discuss further if you wish; just send me a PM if you like.