Uptil now my wife was not working and she and my son are on my mutuelle (I work in the private sector). However, when she starts working in the private sector will she necessarily be dropped from my mutuelle? If so what happens to our son? Whose mutuelle will he be on?
Based on personal experience in France :
Theoretically, your wife may decline the mandatory mutuelle from her employer based on the fact that she is already covered by your own mandatory mutuelle. Notice that the request has to be explicitly done at the time of entering the new company. She might be asked to prove that she is already covered by producing an attestation from your current mutuelle.
(See Peut-on refuser la mutuelle ou la prévoyance de son entreprise ?, Vous avez déjà une mutuelle en tant qu'ayant droit (couverture collective)).
Employers are sometime reluctant to apply this rule.
In any case, your wife and son stay covered by your own mutuelle as long as you keep subscribing to your own employer's mutuelle family option.
Is it possible to subscribe later if one has declined the subscription at the beginning of their contract ? Probably yes but that has to be checked with one's employer.