How common is allergic response to iodine and is it something which one would normally be advised about by a doctor, particularly if your children were susceptible ? When I was a kid a dilute of iodine was a normal thing to apply to a wound.
I do not have statistics for the frequency of iodine allergy. You should definitely discuss potassium iodide pills with your physician before giving them to your children.
Allergies can develop any time, even if previous exposure does not produce a reaction. However, once an allergic response is produced by the body, further exposure to the allergen can increase the severity of the body's response.