Hi Ania, Most families at the British school live this side of Paris - Croissy-sur-Seine, Le Vesinet, Chatou, La Celle Saint Cloud, Bougival. Buses also run from various places (incl. Saint Germain). There is a strong network of "welcome mums" so wherever you end up someone will help you settle in if you want. Socially I have found the school community very welcoming, If you are that way inclined you will not be short of coffee mornings, nights out, companions for exploring paris etc.
I have limited experience but I would say the British school is very similiar to a fairly traditional UK private school -they largely follow UK curriculum for example. I imagine this might help make the transition easier for your children. Whilst most of the children are British there is a very strong international presence, for example my daughter's class has French, Russian, Cypriot, Korean, Jamaican, Australian and Italian children alongside the Brits.
My eldest was Year 1 last year and thoroughly enjoyed it and I was pleased with her progress.
With three children and a dog we decided city centre living was not for us either. We have ended up in La Celle Saint Cloud and are very happy here, for us it has a good balance of reasonably spacious house and garden, woods & parks to walk in but shops close too. It only takes 20mins or so to drive to Versailles (or Saint Germain en Laye!) or you can get the Train and be in Central Paris in 30mins or so.
If you have any questions I'm very happy to do my best to answer!