Ah, a fellow Baxter fan. I've read those books too (though I see after some quick research that I need to get
Phase Space to really complete my collection of the Manifold series). My game actually features elements drawn from them, and
The Time Ships too; for example, the portal to the higher dimensions that my players are partly trying to help open is powered by a Ring about 10 million light-years wide crafted entirely of cosmic string (sound familiar?).
I'd place the Xeelee on the level of Cosmic entities, probably each of them could be considered a Sidereal without Cosmic String (if that makes sense). The Photino Birds would be in the same range, but of course their numbers are far greater. The fact that both races execute plans on a time scale of millions of years says that anything less than Sidereal equivalency is pretty silly.
The Downstreamers, of course, are far greater than that. They'd have to be Eternals at a minimum; the fact that they actually
destroyed a universe specifically to restructure the rest of existence says that much. The question to answer, then, is how high they are relative to things like High Lords; since they are known to have at their disposal an information-storage system that contains everything that ever did or ever could happen, everywhere, in the entire history of the universe, they're guaranteed to have at least one Omnific ability and so need to be at least High Lord (Duad) level I'd say. Of course, given that the aforementioned information system looks suspiciously like the Akashic Records, and while they're using it they're literally the last living things in existence, that pretty much places them above even the Eternal level and into Supernal territory. The interesting question there is, if you presume that multiple consciousnesses exist within the Downstreamer civilization (or what passes for it), what does that make each such consciousness? Is each one an effective Supreme Being? Or is some collective mind they all are part of, the real Supreme Being, and then each mind within it must be an avatar and therefore a High Lord?
Not knowing details of the Last Great Time War, I can't speak to that directly, but I can say that my own game uses the convention that more than one time dimension exists. Parallel timelines, representing the "many worlds" of the Everett-Wheeler interpretation of quantum mechanics, are part and parcel of Reality with everything else. This has the convenient effect of eliminating any possible time paradoxes: for if you go into the past and kill your younger self before you invented time travel, to use one basic example, it doesn't negate the timeline you originally came from (where you the inventor were not in fact killed by your future self and therefore survived to invent time travel) and you simply create a new timeline to follow. This also means that luck has a direct in-game manifestation, since every possible die roll for any given situation is represented in the continuum by its own timeline; thus, when you reroll a die, or make a
Wish to change a result, you're really just traveling along the fifth dimension. Of course, the existence of this fifth dimension also has the effect of rendering time travel dramatically meaningless in the greater scale of things- after all, if making a new timeline changes nothing in the timeline you originally traveled from, then why bother with it at all? Your adventure plots and game style would have to take this factor into account.
Technological singularity, IMO, would best be resolved as an evolution (by the beings undergoing it) to the next higher tier on the chain; thus, mortal -> Deity -> Sidereal -> Eternal -> Supernal. If you want to describe precise game effects for it, and how technological apotheosis differs (if at all) from supernatural apotheosis, then you'll need to decide what each tier of being represents in scientific terms. My own campaign answers the question for Deities and Sidereals; it posits that Deities are beings whose minds are housed in energy patterns rather than physical bodies of any sort, and that Sidereals are minds encoded in patterns written into the quantum foam (or perhaps more precisely, the actual probability waves that govern particle behavior in the quantum sense). I haven't answered the question for Eternals or Supernals; if I'm honest, I probably never will (at least not until we have a definitive theory of quantum gravity in the real world that points to a level of existence more fundamental than the quantum). So getting back to apotheosis, once you know what each tier of being "is" in the scientific sense, you can then explain what the apotheosis process is like when a PC undergoes it- for example, in my games, I invariably described the process of ascension as involving the character's body being blown apart and/or dissolved by some sort of supremely powerful blast or wave of energy, but having his or her
mind remain intact- and then reform the shattered body and equipment from the particles left behind by the energy wave.
In all of this, really, it's wise to keep (Arthur) Clarke's Third Law in mind: "Any sufficiently advanced technology is indistinguishable from magic." The line between fantasy and science fiction- even "high" fantasy and "hard" science fiction- is not as rigid and unyielding as most people believe.