Nut allergy. Maybe. Your children may have inherited genes from you that make them more likely to develop an allergy, but they do not inherit a specific allergy to a food e.g. Nuts. The children have to be exposed to food proteins in the diet, before an allergy can develop. Once one develops an allergy then they are always allergic and need proper medical attention to prevent severe problems.
Possibly. They have a higher risk of allergy than the general population, but not necessarily to the same foods you are allergic to.