So I can't actually find the video in question, but way back before New Vegas was released, there was a video shown at E3 or someplace, which was the first time we got a look at the NCR, showing off their armour, and involved a quest where they led an attack against the Legion in a town which might've been Nipton.

Anyway, I don't recall ever actually finding this quest in-game, so I was wondering, was it cut, and if not, who do I have to talk to to begin it?

Don't hold me to it but I'm pretty sure the video in question was them showing the "Restoring Hope" quest. Which had various things to do but ultimately ending with you and some NCR troopers led by a named Cooper starting an attack on the Legion based in the town of Nelson. Which I can confirm is in the game, you can get the quest in Camp Forlorn Hope.

Actually, I think that's it!
It's not exactly like the one in the preview video, but it does look pretty similar.