First off, hello! This is my first post on the forum (I think, I've lurked a bit).

What happens in the other settings as time moves forward in the Forgotten Realms? I'm talking things like Greyhawk, Dragonlance, Dark Sun, and the setting that provoked this question in me... Planescape.

Is there a unified time that expands over all of the crystal spheres of the Material Plane? Or is it that time moves differently in each of the spheres (like how we obtain updates to certain settings). I'm not even sure if some of the settings or these thoughts are even canon anymore (as the world tree model of 4e was replaced by the Great Wheel in 5e, while also confirming the existence of Sigil and the Outlands... because if that's true then there should be some calculation from events in DR to other calendars across the planes.

I'd also like to know if The Nameless One is canon, or at least will appear perhaps in another module. Seeing him as a mini from Wizkids gave me hope that we will see him sometime.

I’m assuming this isn’t explicitly stated to give DM’s more flexibility. 

Oh, good question! I know some planes of existence are described as timeless in the Dungeon Master's Guide in the section on planes (i.e. the perpetual sunrise/set in the feywild, or the shadowfell's unending gloom), but other than that, I haven't come across any time modules in D&D game publications. Sometimes I'll make reference to time differences with inter-material plane magical portals, such as those that connect cities across great distances; where one city is experiencing early morning, the other city is in the dark of night, or vice versa. But I haven't delved much deeper than that. 

I think there’s a unified timestream.

I've seen a campaign where players left one plane and entered another together but returned to the previous plane at different times (some gone for only a minute or a day and a couple of others missing for a week) as if time affects different creatures differently.

It was made that way for convenience as the players had other obligations for a while, but it did offer a different view of how time relates across planes and created a couple of smaller party hooks and a time-relative hook in the story.

While it made my head hurt to think about it, I like the idea of total flexibility in handling time.

Good point. I guess different planes might treat time differently, but barring temporal magic I think a year on Toril should be the same year on Oerth.

Time generally passes at  the same rate on other planes, with a few exceptions:

1. DMG p.47 "Creatures on the Astral Plane don't age"
2. DMG p.49 Deep Ethereal "it's impossible to measure speed and hard to track the passage of time"
3. DMG p.50, Time Warp "While time seems to pass normally in the Feywild, characters might spend a day there and realize, upon leaving the plane, that less or more time has elapsed everywhere else in the multiverse."
4. In 3.5e, "Time" was a planar trait that could be normal, faster flowing, slower flowing, erratic or timeless (and the Genesis spell could create a demiplane with desired traits).