Hi, we've lived inNyo for almost a year now and have a son almost 3.
There are a few around the area.
For a full list of Childrens facilities http://www.nyon.ch/puce_a_l_oreille/jeunesse.htm
It's an extensive list.
A few others I know about La Nacelle on Route De St-Cergue and Chatterbox (www.thechatterbox.ch)both
next door to each other.
There is a English group called Mops which is on a Tuesday morning around 9:30. It at a church on Rue Juste Olivier. Not far from the train station) There are notices on the door in English for that one.
There is one over on the Fontaines. The name of which escapes me.
I would do a search on Nyon and children on the forum as some of these questions have been answered before.
You will find over the next few weeks these facilities close up for the Summer and things get pretty dead around here.
If you are looking for things to do places to go around the area there plenty around. The local outdoor pool at Colovray is very nice right on the lake. Signal De Bougy is a great place for kids and big kids like me
Anyway if you need any advice about the area or anything just let me know always glad to help. (in other word too much time on my hands right now.)
Enjoy your stay in Nyon.