A key point to remember here is that "After at least 60% of the population will have been vaccinated" is a dependent clause, with "after" being the subordinating conjunction. Therefore, in terms of tense, we do not have to relate the tense in this phrase to our current time, only to the time mentioned in the independent clause.
A statement containing the structure "will have been" is generally not paired with subordinating conjunctions such as "after". Because "after" is a subordinating conjunction, following it with a past tense verb does not have to mean that the verb is describing an event already in the past.
It probably makes this easier to explain if we change the order of the sentence briefly, so as to see what is going on more clearly. If we start with the independent clause, the sentence currently goes "The vaccine will become effective after at least 60% of the population will have been vaccinated".
From here, we can see that it still makes sense and is much less convoluted to say:
"The vaccine will become effective after at least 60% of the population has been vaccinated". Then, we can put it back into your original structure, so that it goes: "After at least 60% of the population has been vaccinated, the vaccination will become effective."
I cannot say with absolute confidence that your version is wrong, but I do know that it is at the very least an uncommon phrasing and the version I have above will be more standard.