It has been hypothesized that the increase in allergic disorders may, in part, be a consequence of changing diet. The primary aim of this study was to assess the associations between occurrence of atopic dermatitis; food allergy; the incidence of wheeze inhaled glucocorticosteroid use in children during the 1st year of life; and cord blood concentrations of copper, zinc, vitamins (A and E), and glutathione peroxidase activity. We evaluated 240 1-year-old children from the Polish Mother and Child Cohort Study. Women were interviewed during pregnancy to collect demographic and socioeconomic data and medical and reproductive history. Exposure to tobacco constituents was assessed based on questionnaire data. At delivery, umbilical cord blood plasma was sampled. One year after the birth, the child's exposure and health status were examined. In the analyses a multivariable model was used. Higher zinc and copper concentrations in cord blood were associated with increased likelihood of wheezing in 1-year-old children. This effect was seen only among children exposed to tobacco smoke at home. We also showed significantly lower activity of glutathione peroxidase enzyme 3 in umbilical cord blood plasma of children with atopic dermatitis during the 1st year of life. There were no significant associations between vitamin A and E concentrations in plasma and children's health. We showed imbalance in the antioxidant defense system in cord blood, which may lead to development of atopic dermatitis or wheezing in infancy. The association between maternal nutrient status during pregnancy and child's health is complex and interacts with other environmental factors such as tobacco exposure. This study was a part of the clinical trial NCT01861548 registered at ClinicalTrials.gov.