Does a specific expiation become incumbent upon a wife who exited the tawaf due to overcrowding and did not complete it, then repeated the 'umrah months later?

Whoever has entered into ihram (definition) for 'Umrah (definition) must complete it, based on the Almighty's saying: "And complete the Hajj and 'Umrah for Allah." In your wife's case, the 'Umrah she performed was her first, and her entering into ihram for the subsequent 'Umrah is not valid because she was still in her first state of ihram. The prohibitions of ihram that she committed out of ignorance before completing the 'Umrah do not require expiation. However, for what she committed knowingly, expiation is due, unless it was sexual intercourse. If it was sexual intercourse, then her first 'Umrah is invalidated, and she must proceed with it and complete it. The subsequent 'Umrah will suffice for her, and she must make up for it with another 'Umrah and sacrifice a sheep to be distributed among the poor of the Haram (definition). As for the children who entered into ihram, the more correct view is that they are not required to complete their rituals because they are not considered accountable, and this is easier for people.