I've heard both that all the recordings were lost, and also that barely any were actually recorded in the first place, which matches up with other reports that the script was never actually finished. So I don't think the data needed to make a complete story was ever created in the first place.
Secret of Vulcan Fury went through at least three complete cast recording sessions (they had to do so many because the script getting mutated because the gameplay was completely broken). However, by the time of the third session, Maurice LaMarche had been brought in to dub for McCoy, as DeForest Kelley's health was failing to the point that he couldn't travel to the studio at that point. But multiple scripts were written for the game, and the game's scenarios, or at least some of them, were indeed mapped out and tested. The problem was that Interplay was relying on its own in-house engine that was doing some entirely radical things, graphics-wise, stuff that really hadn't been done before ... and, as Klingon Academy later proved, Interplay's in-house engines didn't always exactly work correctly.
It's entirely likely that the recordings still exist on a hard drive somewhere in California; Brian Fargo and Rebecca Heineman were both very big on archiving stuff before "oh, shit, we need to save everything" became A Thing in video gaming, although Fargo was ... not particularly good at it

. Whether or not those recordings exist
in a usable format, however, is another question entirely, particularly given how specialized audio recording equipment, hardware and software were at that time: It wasn't just a matter of putting on a headset and laying down all your dialogue in MP3 or WAV format back then and having a perfect digital master on a hard drive somewhere. It's entirely possible that the hardware needed to salvage those recordings would only exist in the hands of a few very specialized collectors anymore.
FC used the first Unreal engine, IIRC, and I remember reading that they had trouble with it.
First Contact didn't just use Unreal 1 (which was a broken piece of shit on its best day), it was Microprose's
very first attempt at making a game on Unreal 1, so it was really probably doomed from inception. They did manage to hack Unreal 1 enough to turn Klingon Honor Guard into being ... well, I wouldn't ever call it
good, but it was at least
playable.