When we are discussing about the allergy to milk, we refer to an immune response which is principally related to the milk protein. Lactose intolerance, is another question, and it is related to the lack of lactase enzyme which precludes the digestion of this milk sugar.
So, we have several proteins in the milk, which are classified in caseins and whey proteins. Each one of these, contains different proteins: alpha-s1, alpha-s2, beta-casein, k-casein, beta-lactoglobulin, alpha-lactalbumin, etc.
Normally the allergy to milk proteins affects the infants which may be allergic to one or more milk proteins.
For my direct experience, I can tell you that several times we have managed this allergy completely eliminating cow’s milk consumption, replacing it with goat's milk.
For more information on this subject, please see this link which should be useful:
Mother milk contains no B lactoglobulin ( the main whey protein in cow s milk) if the infant consumed cow milk the allergy occurs because presence of B lactoglobulin. Mother milk contains alpha lactalbumin but not B lactoglobulin
The allergy to milk may be due to casein as well as whey proteins. Lactose intolerance is an altogether different problem. There is need to completely avoid the milk. The milk allergy may be recorded in children below one year and generally the allergy my cure completely by 10 yrs of age. In emergency condition due to accidental exposure the antihistamines and administration epinephrine may be required to prevent the anaphylaxis. Best you can do is avoid exposure to milk products.