RealityCheque's idea would be really cool. It might turn some of the fluff onto its head, but the story line would still be absolutely awesome.
Ultimately though, I'm casting my lot with the Orks. Yep, Orks.
Nids, while they're certainly numerous and quite lethal, don't come back quite like the Orks do. Orks are a fungus, and whenever one of them dies in battle, he releases hundreds upon hundreds of spores. Not all of them reach maturity, but those that do will present a fighting Ork within a matter of days or a week or two (can't remember their exact fluff). Now, they might be "Savage Orks", without the almighty Dakka or too much technology, but if a Mek is born, he'll stick together whatever he can get his hands on. The result is that as soon as the Nids hit a planet with Orks on it, they're going to be in for a long, long fight. And Orks don't usually surrender.
Necrons are cool and all, but they won't wake up in the middle of a Tyranid invasion. Their gods will feed off the fear caused by the Nid tidal wave, and if they were to wake up their warriors, they would either be eaten by nids, or awaken to find that hey- there's not too much left out there to enslave. So I think Crons would save their breath.
Eldar would retreat to the webway, atleast while the Nids are around. That means that there would be a huge battle between the unified Craftworlds and the Dark Eldar. Now, if the Dark Eldar are anything like Dark Elves, they'll just waste the entire webway out of maliciousness, and not really care who dies- as long as lots and lots of Eldar kick the bucket too. Even if just one of the Eldar factions were to "win", they would probably die a slow death in exile in their Webway.
Without humans to create emotion, and the Eldar all but gone, Chaos pretty much fizzles out. The Warp still exists, but the Daemons and Chaos Gods are gone. Chaos Marines (who stayed in the warp and didn't run out to be eaten by nids) will be safe, almost as safe as the Eldar in the Webway infact, as I doubt that the Nids would really bother with the Warp. The Warp in general would become a safe-haven, which surviving humans might try to get through but hey: if they're alive, the Chaos Gods are alive too...
If GW really wanted to keep it's fluff moving, they'd likely handle it all like this:
1. Nid main fleet appears, bringing wrack and ruin to the world. The Tau are annihilated, and Calgar packs up his chapter and Guilliman and GTFOs for the galactic center.
2. Eldar, seeing what's going on, retreat to the Webway and duke it out with their kin. The winner here does matter, as they'll likely never leave the webway again until the Big Gribblies are gone.
3. Nid fleet begins hitting Ork infested areas. The Orks realize that this is the greatest battle of all time, and rally together in a massive, unified Waagh! Nids are pretty well halted in this area, and splinter fleets (still huge of course) start roaming the galaxy.
4. Mankind suffers terribly, and perhaps even the Emperor meets his end. Now- we're assuming that he doesn't ressurrect in the warp and beat the ever-loving **** out of the chaos gods, because that would annihilate an entire faction... so instead, Mankind covers the whole thing up, but now has no real leadership.
5. With the Emperor out of the way, but mankind threatened by extinction, the Chaos Gods decide that they should probably do something about all the Big Gribblies running amok in the galaxy, so for a brief period, you see a relative "truce" between Chaos and the Imperium, as they both begin fighting to save humanity. This might even bring some of the excommunicated chapters back into the fold (Deathguard).
6. After Chaos and the remaining Imperium have kicked back the Nids, Guilliman realizes that much of Mankind has claimed Chaos as their saviours. This sparks a fresh crusade, with Guilliman declared somewhat like the Pope for the Emperor (who is of course secretly dead).
7. Now the fluff is back to normal- the Eldar are making small raids out of the Webway, the Orks have banded together in one area but quickly dissolve. The Tau are well... their a mistake anyways. Chaos and the Imperium are at it again, Nids are roaming around in splinter-fleets, and Necrons are stretching and yawning and preparing to start the whole thing over again, except with Everybody vs. the Necrons.