Both these two guys wanted to kill a nation and build a better one in its place. Let's place a descendant of the Vault Dweller, the Chosen One, the Initiate,(or descendant if deceased, I don't know the time period of this game), the Lone Wanderer and the Courier all in one room.

These men or ladies or people could become a dream team of super killers and build a nation. Buy a crap ton of ammunition and wipe out an entire state and build your own country!

While this will never happen in a game, wouldn't it be awesome hypothetically if these people could meet up by some twist of fate and actually work together?

Blargh, probably should have uploaded to the Universe...

Well NV takes place in 2281(ish), that's only four years after FO4 and nearly 40 after FO2. So depending on the Chosen One's age in FO2 he can very much still be alive by the time NV rakes place. And both NV and FO2 take place in the West Coast (and considering the Courier's occupation) at least those two can meet up. Don't know how you can fit the LW there though.

There has never been a protagonist named the Initiate in a canon Fallout game.

And this topic seems pointless to me.

The Chosen One is a descendent of the Vault Dweller, right?

Aye, the grandson. And Mr. Bishop, mentioned in Fallout New Vegas and in the ending sliders of Fallout 2, is the child of the Chosen One.