It is common to use the present perfect to refer to something which has happened before now or which has never happened before now:
They have met.
They have never met.
They have been introduced to one another.
They have never been introduced to one another.
There is a nuanced difference between "in the past" and "before now" even though in a strictly chronological sense (where there is no concept of "now" or "the present") the two phrases are equivalent, but that is because the complexity of time is reduced to the simplicity of a timeline by chronology. However, natural language does not reduce time to a timeline.
The present perfect can be used to express the notion of "relevant to the present".
This will be the first time they have met.
"This will be" refers to the situation as something near at hand or impending (because of "this") and "they have met" refers to the meeting as "consummated", as having taken place. Soon it will be something that has taken place.
If you contemplate what soon means, it will become clear that the "near future" is understood in relation to "now".
If you were describing the TV show episode for the schedule page of a streaming service, a terse description might say:
Zach meets his dad for the first time.
You are describing what happens in the episode. That description is divorced from the "now". It is about the episode's plot, and it is not spoken as if it is coming from the mouth of someone who is living in the fictional space-time continuum of the episode.
Your example with the double future :
It will be the first time Zach will meet his father.
strikes my ear as unidiomatic. The subordinate clause [Zack ... father] shouldn't itself be in the future but in the present or present perfect.
P.S. But that double future isn't ungrammatical. However, it seems to have foreknowledge of future meetings between Zach and his father, despite the fact that you only need to know their history to know it's their first meeting. Even so, it seems to suggest there will be a second and a third meeting, and so on, and so it doesn't hit the nail squarely on the head that they have never met before.