That's because it was explicitly stated in the game. I couldn't care less about what and when or why, because FFX-2 was a really crappy game with plot holes, and very little symbolism beyond "look at us, we made this game to attract people into playing it". Biggest waste of money in my life. Even worse than that dollar I paid my friend to do my Trig homework for me that one time. Every single answer was... correct. Not my style of work, obviously.

(SPOILER)
And Sin did appear before the end of the war; just not that long did it have between him destroying Zanarkand and ending the war. Shuyin escaped the battle anyways, and Shuyin snuck into Bevelle, as did Lenne, which explains how Shuyin was to use the weapon. The war was either already over, or SO shortly before it ended, it would not make a difference. I doubt they wanted to get Yevon's signature and say "you lost, now sign it to prove that you admit defeat".

They escaped, and they fled, and Shuyin had the plan to use Vegnagun to destroy the people of Bevelle. Remember, that from the first appearance, there is a bit of time before people can react. Considering the size of the war, and the technological evidence from Bevelle, it's not surprising they could build Vegnagun in such a short time. But since they died/shot, Shuyin could not wake Vegnagun. The FMVs of him talking to Vegnagun were probably not far from each other in length, maybe a couple of days or less.