It is rude and would annoy me. If I fired off a reply right away it would probably be a little nasty, but if I gave myself time to cool off a bit I would just let it go.
When is/was the bday? Did he/she at least apologize or acknowledge the delay? If so, I would tell them “no problem, maybe we can do something next year.” If not, I would say that “thanks for the reply but the child’s bday has passed.”