If you were in the general proximity to the nearest star to Earth (besides the Sun) and you saw it turn to a neutron star or black hole at the very end of it's star cycle, how much longer would it take the people on Earth to see it? -please feel free to correct me on any of these phrases I might have misused.
If by "general proximity" you mean "reasonably close", i.e. a few light minutes away (Earth is 8 light minutes away from the Sun), then people on Earth would see the star evolve in the future, the time dictated by how far away the star is.
Our closest neighboring star (besides the Sun) is Alpha Centauri, around 4.37 light years away from Earth, so if it turned into a neutron star or something else right now, we wouldn't notice that until January 2020. A similar reasoning can be applied to any star in our galaxy (extragalactical time measurement is a bit more complicated because you have to put Hubble's law, i.e. the expansion of the Universe, into view).
Note, however, that all three stars in the Alpha Centauri complex aren't really material for a spectacle, unlike Betelguese in Orion (which may even explode during our lifetime), or some other examples I can't think of currently.
No one ever sees a black hole form without going inside.
Instead you see something evolve into something that looks more and more like a black hole. And eventually it looks so similar than if a friend casually asked about it you might tell your friend it is a black hole because the difference hardly matters to your friend. But you'd know that you haven't seen it yet.
In particular, reactions that normally take place during shorter and shorter time intervals would be forever newly visible to you. So when your funding agency asks if you need more money to watch the star transition you'd always say yes since there is new physics to see. The physics of the very fast and high frequency slowed down and red shifted for easy observation. Which very much is happening before any black hole forms.
If it just forms a neutron star, then if earth sends a "is it a neutron star yet, the time here is xx:yy:zz" every second then they will wait around 9 years before the time stamp they send out is sent back with a "yes it it became a neutron star when I got your xx:yy:zz message." And it takes about nine years since that is about how long a round trip light signal takes to get near the nearest non sun star and back.