Firelink shrine Hub is definitely after Untended Graves shrine, time-wise. You can tell by Gundyr. I just read the post on Reddit it's covered in which sources the relevant descriptions.
https://www.reddit.com/r/DarkSouls3JPN/comments/4e99hj/lets_talk_lore_8_spoilers/
Good catch on making a connection between the coiled sword at the start of Lothric High Wall and the fragment in UG shrine though. I don't think there's any reason to assume UG Shrine is separate from the rest of the world. Makes much more sense to assume that Hub Shrine is, due to the timeline discrepancy and the fact that the sky stays the same there, no darksign sun. I mean, I use "sense" loosely here, mind you.
UG Shrine / Abyss connections... tempting. Artorias; armor, perpetual dark... but other than that there isn't anything to go on. No black goop monsters, no humanity, a bunch of black knights... I don't think so, personally.
So timeline: Untended Graves Shrine and rest of the game > Firelink Shrine Hub > Futurelink / Kiln of the First Flame.
Don't ask me how or why but that's the only thing that even remotely makes sense.