There's a rather large Anglo-Saxon expat community around Nyon, so you'll meet lots of people who do speak English.
In regards to day-care, places can be difficult to find.
Availability (when there is place) tends to depend on whether the mother is working or not.
I'd suggest to look up the American Women's Club in Geneva as they do have all kinds of activities for moms with little kids (playgroups) and they also offer courses to help new expats adapt.
Just to clarify, what specifically are you looking for? Daycare is generally for children below school age, as your daughter is 5, being in school is the norm and in fact required at this age. Are you looking for after school activities, occasional babysitter, or something else? I can recommend an excellent creche (daycare) but it would seem very inappropriate for a 5 year old.
At 5, she should be in enfantine in the mornings, whatever suits you and her in the afternoon (nanny, jardin, sitter, creche, playgroup..). I wouldn't go for Champitet, eventhough there are people very happy with it, I think if you want to stay here longer and have your child integrated, speaking local language, with buddies to play with within the area, keep her in local school. Plus, the cash? Hello? I don't think what you get corresponds with the cost. Most people drive their kids there to private places. Local schools might be harder if you don't speak the lingo, at first, but in the long run, there is nothing nicer than to be able to schedule playdates without major driving and organising, just bumping into buddies casually will make your child feel at home a lot quicker.
If you use the search feature here, you will find some threads on Champittet. In sum, I think the Nyon school has a lot of problems/issues and would not recommend it.
I think you would need to decide if you want a public or a private school, then go from there, ie.which curriculum, which languages, what values, what approach to learning, etc. If you decide on a private school, there are much better choices than Champittet in the area. If you decide on the public school, then the local school is the choice.
Have you tried the Migros Ecole Club - www.ecole-club.ch - lots of interesting classes, including Zumba, some of which even fit in with the school timetable. Can't offer any advice on gyms (except that I've heard that a lot of them round here are expensive, dirty and unfriendly!) but there are some good dance schools in Nyon as well as yoga and pilates studios. A quick search on www.local.ch will give you the details.
Regarding childcare, maybe your child's school is part of the Reseau Nyonnais d'accueil de jour des enfants - which offers lunches and afterschool care to kids. See http://www.nyon.ch/fr/vivre/enfance-...fants-939-5419
[I find Google Translate works well if you need a translation]