If this were to happen the infant would most likely be prone to more frequent and serious infections because the thymus gland plays a role in the defense mechanisms of the body. Other immunologic organs like the spleen and groups of tonsils would attempt to protect the host.
Newborn infants have an immature immunologic system that develops over months and years. All newborns are more susceptible to infection for this reason. Some immunities are gained from the mother at birth, others are received from the mothers milk (breastfeeding), and still others are received from coming into contact with an offending pathogen. Childhood immunizations also tremendously help to fight off invading pathogenic microorganisms.