Well, I'd think that a photon torpedo would be more likely to sport warp tech than anything, but, since they usually avoid that unless they are already traveling at warp, I hesitate to immediately make that assumption. I looked at the linked video, and I think in this case the first shot against the Cardassian warship actually took a few seconds.
At about 0:33/0:34 on the video, we hear a chime that sounds a bit like a digital phone ringing, immediately followed by Data stating the Phoenix was firing photon torpedoes. After he finishes his sentence, he looks up at the screen, there is a long pause, and they cut to the display at about 0:38.
At this point, the Cardassian ship is barely within the blue rings that indicate the Phoenix's weapon range, but I'd say the ordinance was out of the tubes at least a few seconds, possibly as many as 5 seconds, from the time we hear the chime that seems to indicate a weapon launch.
The same "ring tone" is heard again at about 0:56/0:57, to which Data immediately says that Phoenix was firing again, only this time he doesn't even manage to finish his sentence before target was destroyed, about 2 seconds later. This time, the Cardassian ship was closer, I'd guess just a tad past 2/3 the reach of the weapon overlay. Assuming both attacks were from torpedoes, Id take that to be about 100,000 kps, or 1/3 c. Still pretty darn quick. Now if only they moved that fast when we were actually
looking. :D
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EDIT
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Ugh, working off a vague memory of that page, I skipped over it and took a quick review of the original video, only to have a weird gnawing feeling send me back later.
Then I discover that I basically rehashed the exact same ground, but not nearly as thoroughly. I even forgot the ships were moving!!! Now I just feel like an idiot. There, now I said it, I can move on....