Private schools will vary, but probably around 20k to 30k a year.
To be honest a quick google doesn’t show many private schools at all. One I found only goes up to age 12 and the other is Catholic run.
There would be costs for lunch and after school care if you needed that.
There are private Montessori schools for that age group and the Ecole Catholique which is private but follows the Swiss curriculum for the canton. Both of these options provide full day options and are popular with working parents.
As for the older child it will depend on what sort of school you are looking for.
As things stand at the moment there is no international English speaking school in Neuchâtel for that age so if you want your child to have their schooling in English the nearest schools would be in either Lausanne or Bern. There were plans to open one here but I think Covid might have messed up those plans a bit.
There is a private secondary school called Ecole Moderne which does offer some sort of bilingual programme but I don’t really know how it works. The fees are around 20000chf per year I think.
There are very good secondary schools here and they will provide extra language support for children who don’t have a good level of french.
There are good schools available for free all over the Canton. Posts crossed BM.