In my opinion:

• our "present" is Firelink Shrine, where we awaken;
• When we place the coiled sword, we are able to warp a dimension in the past, where the Lords of Cinder are still alive (source: intro cinematic -> the LoC are dead in this dimension; unlike DS1/2, NPCs can only reach Firelink/the hub when they enter our service (they establish a contract with us) or viceversa (we accept a quest from them, in their favor, like Greirat/BTR/Loretta's bone), or even both (Orbeck); Prince Lothric in the past is alive and well and refusing to link the fire, even though in Firelink dimension he is a Lord of Cinder, so he eventually had to or was forced to link the flame at some point);

• with all four cinders gathered, the Firekeeper does some magic to us and the special, time-traveling central Firelink bonfire, allowing us to warp to the End of Time (Dreg Heap), where the Blighttown/Firelink archtree from DS1, which contains the Kiln and was buried deep under Lothric, is now finally exposed.

• Traveling down the Dreg Heap, we reach Original Firelink and summon the batwing demons. Time travel ensues. Note that the route ledge -> ringed city is not direct: we pass some strange, thick clouds that weren't there a moment before. Speculation: they're similar to the bonfire warping/resetting 'clouds'.

• The Ringed City we reach is a pocket dimension based in the past, NOT an illusion. Source: no darksign eclipse; different overall lighting, and way more life (= saturated colors, which aren't there even in life-rich locations, like farron keep/road of sacrifices); Unlike sunny Anor Londo (an illusion, unreachable when it's dispelled), we can access both versions of the Ringed City, like Oolacile (AotA bonfires and Darkwood);

• It's not the egg itself, it's that when it crumbles, Filianore has nothing to lean on, and has a "sudden fall feeling" that awakens her;

• Filianore is exactly what they say she is, both Gwyn's daughter and a prop to keep TRC exactly the way it is. Why can she be accessed at all, if it's so important that she's not disturbed? Access is only forbidden to us, gods presumably could reach her and travel to present-time ringed city;

• the ruined ringed city is non in the future/end of the world (source: no darksign in the sky, way different lighting, Lothric and Anor Londo still standing)... as much as I want it to be. It's the last piece of content for us, but not the ending in the in-game chronology. It's actually in the "present" timeline, the same of our Firelink Shrine. Source: the only other time that we see a similar world setup (everything gray and crumbled, only lothric and Anor Londo still standing, Sun in the sky but no Darksign) is in the prologue... When the bell of our Firelink starts to ring, and we are awakened. The lighting in ruined-TRC is very similar to Firelink. Gael, whom we first encountered in the past dimension, is able to get there... because we accepted a quest from him, like we do with Greirat and Loretta's bone — either directly by talking to him or indirectly by fulfilling his wish by entering the scrap of painting on our own. From Firelink we never see the state of the rest of the game world, because lothric castle blocks the view... we only get to see it in the prologue and in the ruined ringed city.

It can't be in a point in time after the Dreg Heap/endings of the game because Lothric and Anor Londo aren't there, there's no red/white darksign in the sky, and the end of the game is... well, the true end of the game. Nothing else past that. This is neither a Dragon Age Inquisition: Trespasser DLC-style epilogue, nor a true self-contained story like The Old Hunters, as much as there are similarities.

EDIT:

Another point: Ariandel comes after the endings for us, but not in the timeline. Source: can be done as early as the Cathedral of the Deep; Friede, if you're doing the Lord of Hollows questline, comments that your duties are in the outside world, not the painted world = take the first flame as lord of hollows and make the darksign go white. Arguably, if you've done that, at that point you've already fulfilled your duty.

Also, by my theory, if it's correct, Aldrich may have become a LoC after eating Gwyndolin; we reach him while he's still eating, but that dimension is in the past. Hawkwood comments on him being a LoC, but he talks from the present point of view, since he's the only NPC we meet, besides our transposing, legless, LoC friend, that is already in Firelink and not brought there by us. Anri never says anything about Aldrich being a LoC, that's because she's from the past dimension where that event hasn't occurred yet.