My thought is that the Temporal Wars occurred everywhere, all at once, with the anchor point - the trigger in a sense - in the 30th century.
Basically, I think we need to take a step back and observe Federation history at a "gods eye view" level.
Almost all of Star Trek we've seen up to this point, was in the first 250 years of the Federation (and pre-Federation time), spanning approximately 2155 to 2399. This is nothing. It's only slightly older than the United States in the Present day. Across human history, here today, gone tomorrow powers reigned for a few hundred years, and then were gone. The Roman Republic lasted about twice that time span. So in a sense, all of Star Trek we've seen, from the Coalition of Planets, to the Romulan War, to the Klingon conflicts, to the Dominion War were all the major events of still, even then, the Birth of the Federation. What the Federation was in 2375, for example, is but a seed, a foreshadowing of what it would become, no more than how a Roman Republic that ruled just the Italian Penninsula and modern day Spain was a foreshadowing of the Empire that would emerge centuries later.
I think Daniels in Enterprise by vaguely describing just how different the far future was handled it gracefully. His Earth, his Federation are decedents of the Federation of Archer, Kirk and Picard, but are also in its own way distinct, flavored by a 1000 years of history and expansion and change.
I think the Dominion War, in a historical context, could be seen as an analog for the Punic Wars. The Federation had major conflicts, but most were quite short and comparatively limited in scope. But the Dominion War was by far the largest conflict in modern galactic history, involving tens of thousands of ships, billions of lives, and every known power in two quadrants. But it was also the first time that the Federation, after expanding for over 200 years and it its then zenith,, came in conflict with an adversary, ancient in nature, from far away that in many ways was superior to it. This is directly analogous to the Roman-Carthengian conflict. Carthengian power predated Rome and controlled the Western Mediterranean. But Roman expansion inevitably brought it into direct conflict with Carthage. Both could not co-exist in the region in a sustainable status quo. Both were diametrically opposed expansionist powers. The Punic Wars were inevitable.
In the same way, so was the Dominion War. As the Federation spread into the Alpha and Beta quadrants, time and expansion made it mightier than the Romulans, Klingons, Cardassians, Gorn and Tholians, all one after another. If it were to expand into new quadrants and unexplored sectors of space - far away lands similar to North Africa of Roman Republic period - it was going to come into conflict with great powers.
And something was notable in that. Recall Sloan's prediction for Bashir - after the Dominion War, the Klingons would take a generation to rebuild, Cardassia would be occupied, and the great powers in conflict would then be the Federation and Romulans. But the Supernova of 2387 essentially destroyed the Romulan Empire. This left the Federation alone, recently victorious in it's war against the greatest empire of another quadrant, as a great power in all of local space.
The Dominion War, in a sense, "made" the Federation what it would become in every bit the Punic Wars did for Rome. The Punic Wars paved the way to Rome expanding into all the Mediterranean Basin, which it did with little resistence. In my view, the Dominion War, combined with "luck" in the form of the Supernova of 2387, did exactly the thing for the Federation. The Federation really had no rivals and limitations to expansion anymore. In the 25th or 26th century century it would add the Klingons. It would grow and grow in the Alpha and Beta Quadrant. It would have to go all the way to the far distant reaches of the Beta Quadrant and into the Delta Quadrant to come up against its first known true rival - the Borg. And in the other direction, it would need to go comparatively farther, deep into the Gamma Quadrant, to find Dominion space (supposedly vast). The Dominon in the early 2370s expected the Federation (which they knew about ahead of time) to reach them in about 200 years wormhole aside, or 2570. That should give you a sense of what they figured the rate of expansion of the the Federation would be. But the Federation won the war against the Dominion, and one could suppose that, 200 years later, the Dominion would face a Federation on a completely different level of power.
My thought? The Dominion of the 26th century was basically the Federation's analog of what Persia was to the Roman Empire - the wall to further expansion, one that would eventually be overcome. A great power that could go toe to toe and fight exhausting conflicts at regular intervals, but one that was doomed to loose against the greater power. Perhaps that's what the "Founders Fourth Homeworld" meant. Perhaps the Federation just kept going to the point it drove the Dominion into the fringes of the galaxy, if it continued to exist at all.
So there were are, in the 30th century, with 350 "planets'. But remember, the Federation uses "planets" in different ways. Sometimes litterally, but usually it means polities. All of the Klingon Empire, for example, could be one "planet", akin to a Federation version of Texas. Just one big political subcomponent. With 350 planets, the Federation could control almost all of the spacefairing galaxy, with that vast distance travable by next generation warp drive or quantum slipstream (which Book's ship has) In fact, judging by the dilithium shortage, I bet that's exactly what happend. The Federation expanded everywhere.
So what was the Temporal War? The greatest conflict the Federation ever fought. The one that was always coming for it. It expanded so far, and saw down all rivals. It, in many ways, became the galaxy. And the only way to stop an adversary like that is via an asymetric attack. And that's what I think happened - the Federation's rivals of the 30th century era, mostly composed of species we don't know yet and not encountered by the late 24th century, decided to attack the Federation's existence throughout history and recruit allies across the timeline who wanted to stop the Federation from expansion (expansion that, by the 30th century, had already occurred). In a galaxy without the Temporal War, there is no reason the Federation couldn't have expanded forever, and become everything, everywhere, within our Galaxy and maybe one day beyond.
I think the Temporal War was fought almost on abstract terms. It didn't really involve ships and torpedoes', but agents and technologies to both "lock in" events and "cancel out" events... time manipulation technology that would be difficult to capture on screen. Let me put it like this, if the Dominion War was fought with the 24th century equivalent of chariots with bows and arrows, then the Temporal War was fought with thermonuclear weapons. It was a "higher form of war".
And it's really not surprising that the conflict, though victorious, basically knocked the Federation down flat, and allowed a slide into decline that predated the Burn (as the President of Ni'Var stated). It took everything the Federation had to just continue existing. The Federation by the 30th century was no longer an young entity, but an ancient one. And the Temporal War was its equivalent of a World War. The most expansive, expensive and dangerous conflict it ever fought, on a plane that even people of Picards era couldn't grasp.
This also has a history analog. After the division of the Roman Empire, the Eastern Empire (Byzantines) continued the campaigns against the Persians. And in time the Byzantines won, but the effort depleted the treasury, exhausted its resources and depopulated vast swaps of territory. It left the Byzantines and conquered Persia vulnerable to a new adversary from a land that had been given little though - the depths and vastness of the Arabian Peninsula, where Mohammed's Caliphate arose. First they swept through conquered Persia, in suzentry to Constantinople. And then they seized all of the Byzantine's holdings in North Africa, the Levant and most of Asia Minor. The Byzantines (and probably Persia) at it's height, could have easily withstood the caliphate. But given that they had ruined each other, at that time, they could not. And it was all the Byzantines could do to not fall themselves.
I think that's basically what the Temporal War was. The win condition was the Federation continued much as it was, but exhausted from it. Somewhat declined. And then pushed over a cliff and by the 32nd century, the galaxy is a lawless free-for-all of marauding armies and petty kingdoms again, much as it within just the small slice of the Alpha Quadrant in Archer's time before the Coalition of Planets was born. I think for the long time fan, that's what the special emphasis on planetary defenses (shown 3 distinct times) in Discovery is supposed to be: a space-based equivalent of walled cities. When Rome was in its infancy, walled cities were common. When the Empire grew and the frontiers pushed far away, cities out grew their walls and the walls were torn down. When the empire decline and fell, new walls went back up, because the legion couldn't defend the cities at the front. Those walls persisted in Europe for a millennium until cannon made them obsolete. In the 32nd century, without Starfleet, the "walls" of the 22nd century went back up, more or less. That is another legacy of the Temporal War.
This is why I hope that, when Discovery ends, we take a massive leap forward, to like the 53rd century. And we see what has become on the other end of the Federation Dark Age - a Federation that builds Dyson spheres, where citizens live for 500+ years, has colonized all the dwarf galaxies around the Milky Way and is starting to explore Andromeda and Triangulum. Where its very citizens are a synthesis of biology and programmable matter and information. I would love to see the Federation evolve into essentially an abstract entity operating on a different plane, and the Temporal War, in that context, every bit the trigger for transformation to the next stage of its evolution, every bit the Dominion War was to the Federation that, as Q so keenly observed, was so quaintly preoccupied with the Romans and Klingons and charting nebulas for a few hundred years, as if in the big scheme of things, that matter. The Federation not yet Gods or Q, but as countless superpowerful races encountered in the 23th and 24th century looked out to the future for, would be by that time a power that could be talked to in non-child like terms.
