Our son was diagnosed at 8 weeks while I was EBF'ing - symptoms were constant crying, fussy when feeding - screaming for more, then screaming when feeding - choking on milk, projectile vomiting, 8-10 liquid poos per day, poor weight gain (from 75th to 25th), wouldn't lie flat, constant gasping, coughing, congestion after feeding, hives on chest and forehead, cradle cap covering whole head.
I cut out dairy and soya and symptoms started to improve after 4-6 weeks, but didn't really improve fully until we put him on Neocate, probably because he has since tested positive for egg and peanut as well, so may have still been getting traces of them through my milk.
By the way, it's not as clear-cut as saying that if it's an immediate reaction then it's an allergy and if it's delayed it's an intolerance. Our son has a delayed onset allergy to dairy and soya, but an immediate allergy to egg and peanut. A delayed-onset allergy is just that - it doesn't mean he can tolerate dairy, the immune system will respond to the allergen in the same way as someone who has an immediate reaction, it just means the reaction itself (upset stomach, eczema, hives, vomiting, all of which would get worse with more exposure) might not happen for anything up to 3 days afterwards.