What if the augments had wiped out regular humans in the eugenics war and had taken their place? I imagine they would have discovered warp drives sooner. For sake of argument take the normal first encounter with the vulcans that humans experienced but substitute them with augments. What is an augment led federation like?
As much as I hate agreeing with Sollvax, this. The Eugenics Wars weren't just between normal humans and augmented ones, they were also between regional augment warlords, who, consumed by aggression and arrogance and with little sense of morality or self-improvement outside of martial strength, constantly squabbled among themselves for dominance. With the world to themselves, they would only continue to do so.
Remember, with the extinction of normal humans you're talking about a scenario where the surviving population of the world is in the mere thousands. The wars only consumed 30 million people at a time when the world's population was 6-8 billion (depending on whether you go by TOS or Enterprise's timeframe for the war). Devastating, but not exactly a major depopulation - 0.5% in the worst case. WWII killed 70 million when the population was only 2 billion (3%) and WWIII killed 600 million when the population was 9 billion (7%).
If it is the 1990s Eugenics Wars of Trek Lore, then I agree with Solvax that they would become extinct. I think this would happen due to in-fighting and wipe themselves out way before they discover FTL travel.
However, occurring under circumstances like in Andromeda, when Dr Paul Musevini left Earth to create a genetically superior species, I can see that being a plausible outcome as well (Bearing in mind that Nietzscheans were equally on the verge of wiping themselves out by infighting, but had enough space (no pun intended) to spread out enough to not be as 'on top of each other' as Earth-based 90s Augments would be...