Is there a relationship between speech delay and food allergies?

No. People blame "food allergies" for many things. Allergic reactions to foods include itching in the mouth while eating the food, followed by rashes or hives within 5 minutes to four hours of eating the food, and/or nausea, vomiting, abdominal cramping, bloating, and diarrhea. Severe reactions can include wheezing, low blood pressure, and death. Other things, like autism are speech delay, aren't.