"Overstay" is to stay (physically) longer than allowed, e.g. to overstay one's visa is to stay in the country even after one's visa has expired. In medical contexts, "delay" is often used. "If you delay your child's vaccine for [over] two months, you'll have to pay a fine."

Thanks for the reply, Mika! You are right saying "This means that it had needed to be done a while in the past but you haven't had it done yet." That's exactly what I wanted to say. I used Russian - English dictionary to come across "overstay". It appears to be definitely wrong.

What do you mean by this sentence? You can have a vaccination (vaccine) which is overdue. This means that it had needed to be done a while in the past but you haven't had it done yet. You can, however, overstay a welcome which means that you were supposed to leave but have continued to stay when you are not wanted. If you explain what you mean, I could help you find the right terminology :)

No, es mejor decir, no lleve a vacunar a mi hijo el día que corresponde.