Like, would someone sitting in Granada, Spain in the 1500s be aware that while he's experiencing an evening with sunlight, in India or China it might be nighttime for people there at the same time.
This depends how educated this person was.
Scientists knew (or lets say strongly assumed, hard evidence came later) since the ancient Greeks that the earth is round. Pythagoras was the first who discovered that, around 500 BC.
Others provided more evidence, like Aristoteles. This idea was a little forgotten again for a while, but through this whole time there were people who believed the earth is round.
Copernicus and Galileo around 1600 really got into it again, and provided more evidence.
So anyways. Since all this was on the table on and off from 500 BC, it's safe to assume that it was also clear that when it's day in one place, since the earth is round, it must be dark in another.
Though not all people believed that the earth is round, actually in some periods it was considered stupid to believe it.
Hence to answer your question, yes, if you were in the group that had heard of the concept of a round earth and believed it, then you were aware.
Time zones however came much later. Until about 1900, each town set their clocks to the one official clock in town (and that clock was set every day at noon, when the sun was highest). At least that's how it was done in most of the western world.
This worked great, until people started to travel more. Traveling was super messy because every town had their own time lol.
Once the railroads came in the US, an official time was absolutely necessary, and since it's not all the same everywhere with day and night, it needed time zones.
So at very latest from circa 1890-ish, people definitely knew.