Can allergy to shampoo affect lymph nodes on the back of the neck?

Not likely: Allergy to shampoo is of the contact type, and will therefore affect the skin long before it affects the lymph nodes. If you don't have a significant rash all over your scalp (the areas that come into contact with the shampoo), the swollen lymph nodes are probably not related to the shampoo.

Yes: Yes those lymph nodes drain the scalp and they can enlarge with any irritation or infection of the scalp.