Did the Voyager episode Future's End occur in the prime timeline?

In-universe, this is one of those instances where a closed time loop was interfered with and had to be corrected:

Chakotay: And every few years, there's been an equally revolutionary advancement in computers, all from Chronowerx Industries, all based on Starling's crude understanding of 29th century technology.

Janeway: Are you thinking what I'm thinking, Chakotay?

Chakotay: I wish I weren't.

Janeway: The computer age of the late 20th century-

Chakotay: Shouldn't have happened.

Janeway: But it did. And it's part of our history. All because of that timeship.

So no, there was no alternate timeline during the 90s. The anomaly mentioned at the end of Part II was simply Voyager's existence in the past, not the timeship in the 60s - restoring the timeship to the future would've set Earth back decades, maybe centuries, and not led to the timeline we know.

Within canon, an explanation for there not being any Eugenics Wars simply isn't given.

However, stepping outside of canon, authors recognized this problem and came up with a solution of sorts: The Eugenics Wars were a behind-the-scenes fight, more like a chess game with nations, and the general public did not know about the existence of the Augments.

This particular retcon makes it entirely possible that the Eugenics Wars did/are still happening...

To answer the comment on the question:

And don't forget that prime Braxton had no memory of it. This entire episode is kinda contentious, though Richard makes an excellent point that I'd never considered, in Braxton's own ship forking the timeline. I guess that basically solves it. – Lightness Races in Orbit

The original Braxton that went back in time was from further in the future than the Braxton at the end of the episode. That's why he didn't have memories of being in the past. The timeline split, if there was one, was either when Voyager left the 90s, or when Voyager returned in the 24th century:

Their existence in the past caused a younger version of Braxton to go and retrieve them, preventing the temporal explosion from happening in the first place - and splitting the timeline such that the explosion now occurs in a future that is not in the Voyager's future, but is still a possible future of the 1960s and 1990s - hence no paradox occurring.