There aren’t going to be any female Krogan on Earth so Humans will only have the ones still alive after ME3 to venerate—until they die. That would be hundreds of years, of course. The surviving Asari will live as long or longer. Generations of Humans will grow up around the very people who fought what will become a storied war. That can keep it from fading to a myth on Earth, if not anywhere else in the galaxy.
Asari can interbreed with anyone, which means that will eventually start outnumbering the other species, except Humans, and us only because we have a head start with the billions of people who don’t live in the large cities targeted by Reapers. I suspect the Asari would try to take charge of Earth in the future, which would make for an interesting story.
Turians and Quarians will have a very different problem: one of food. Depending on the size of the forces Shepard brought and the mortality rate they faced in battle, they may have an awful lot of people who can’t eat anything that’s grown on Earth.
The Quarian liveships produce food but it isn’t really said how. Do they have soil of some sort that allows them to grow plants or do they live on some synthesized sludge—obviously ingested through their emergency induction ports—that contains all of their required nutrients? There’s no way to know (though that last might explain why they’re so damned anxious to get back their home world) so the possibilities are wide open for speculation about how long they can feed everyone.
If the Turians segregated their fighting forces by gender, as we know the Krogan did and the Salarians seem to have done, the species will die out on Earth in the coming years. The Asari, however, will certainly start having children with all of them, Quarians included. They seem to like a lot of variety in their diets, so to speak.
The Krogan have no Wrex to lead them in the rest of the galaxy but neither do they have a way to conquer other systems because the relays are gone. The fight between Krogans and Turians on Palaven would be an interesting side adventure but it cannot last forever as all of the female Krogan are back on Tuchanka. With luck, Eve can have enough influence on those left on Tuchanka that they let the planet recover as the greenery in the genophage cure mission implies it can, rather than overpopulating it and wrecking the planet for good.
With no relays the various species that came to the Sol system to fight are stuck here. Something like a liveship that produces its own food would still require its crew to commit to generations on the fleet trying to reach the next inhabited system. That would work for the Quarians, who have long practice doing so, but how willing will they be to share their technology and wait around while other species learn to make their own ships?
Then, of course, any expedition leaving for another system must face the question of fuel. With the mass relays, ships needn’t carry much fuel because they crossed inter-cluster distances in the blink of an eye, more or less. To power themselves across those same reaches of space ships would need to carry enough reaction mass to last them the whole trip or bring equipment to mine helium-3 or eezo or whatever along the way.
The Quarians must have that technology, after their lengthy sojourn in space, and the game tells us it exists in the Sol system. With the system itself having been pretty well depleted and a huge fleet now occupying it, fuel is going to become a scarce commodity in quick order. If people want to fuel their ships, as the Quarians and Turians must if they want to eat, they’ll have to start ranging farther afield in the near future.
Salvage rights look to become a major issue. Between the dead Reapers and the destroyed ships of all of the fleets, there will be plenty of detritus and parts floating around, from fuel sprayed into open space to perhaps nearly-whole ships. Here, the Quarians have the advantage of being expert scavengers.
With any luck they can be convinced to share that expertise and help make the most of the remains of the Citadel, the Crucible, the Reapers, and the destroyed mass relay. So much advanced tech just floating around the system with little else to occupy the space-faring folk stranded in-system offers the opportunity to finally understand how it worked.
That could lead to rebuilding a whole lot faster than starting from scratch. One presumes other systems will be attempting to reverse engineer their own relays but they won’t have the makers thereof (assuming it was, in fact, the Reapers that made them in the first place) or the Citadel to provide context and possibly spare parts.
A single working relay won’t do a lot of good without another to catch the ship on the other end. With all of that tech just begging to be adapted, however, the folks stranded in Sol system may be able to boost their own speeds to that evinced by the shockingly-fast Reapers, who zoomed into the galaxy and distributed themselves around it with what seems to be utter defiance of plausibility and physics.
None of that will happen overnight. The interplay between species that would take place over what I imagine will be decades, at minimum, would make for a great story, one that I would love to read. How many Turians will the the Quarians liveships feed and for how long? How quickly will the Asari breed and for how long will they be willing to be a secondary power in the system? Will Turians and Krogan look at cloning or some other means of extending their presence or will the research on inter-system travel move quickly enough to make that question moot?
A great deal would depend on whether AIs and other technology were destroyed in the semi-magical EMP released by the Crucible. In the worst case, the systems controlling the environmental and drive systems of ships would have been eliminated. Those trapped within them might face horrific deaths as they drift, running out of air and unable to land. Or containment of their mass effect drives could fail, in which case their ends would arrive rather more quickly, I would think.
You could find enough questions to fill a whole series of novels. Established lore gives some guidance but leaves more than enough leeway for a truly creative exploration of the politics and grim realities facing the various species left floating around Earth at the end of Mass Effect 3. I wonder if BioWare could convince David Weber to take a crack at it. I don’t know about you, but I’d love to read more.