Given that the Millennial generation is often judged as ending somewhere between 96 - 2001, the best time to see where generation z is going to end up is 2020. At that point, we'll have a handful of years in the 18 - 24 group as being genz, the number of years depending on when you think Millennials end. If they really trended more Republican, the 18-24 group should lean more Republican in 2020 compared to 2016. It's not enough for the Democrat to do a little worse, as even while Clinton performed worse among 18-24 year olds, Trump also performed worse than Romney, which was already pretty low.
Anything before then would be too ambiguous. Personally, I think that even if generation z ends up more conservative than Millennials, it won't be an immediate drop in vote share for Democrats. It would probably be a slow-ish decline, and highly doubtful it starts anytime soon. Republicans need to re-brand to make inroads with young minorities (or massive inroads with young whites) for that to happen. If it does, it'll probably be the tail-end of the genz that gets more Republican-leaning.
I think it'll be pretty hard for the GOP to take the "massive inroads with young whites (specifically males)" route, by going full-on alt-right at least. One of the reasons that the alt-right got so popular as it did was because it was a rebellion from both the political establishment and the cultural norms of our times. Now that the alt-right has essentially BECOME the establishment, and an incompetent and terrible establishment at that, it'll be harder for young whites to be enamored with the alt-right brand. (Which is one way a Trump victory might be better long-term than a Clinton victory; imagine how much the alt-right's popularity would skyrocket if she was the president).