No, honestly... For them there's not a single reason to actually want one of these. Enclave? They want to get rid of mutated freaks, not create more of them. Brotherhood/Outcasts? What for.. ? Well maybe to extract radioactive material from them to use as power source.... That would be only reasonable usage for them. Supermutants? Seriously, imagine these big greenish guys standing next to the nukes. Effect: "wut is dat big rund things?". From our point of view, having a nuclear arsenal might be a wish of every military faction. But from the point of view of such factions living in a world burnt down to crips by these things - they all just want to stay hell away from them.

The Megaton nuke could have easily been from an American bomber that crashed. After all, if indeed the Chinese did make first strike, then not all of the planes may have gotten far by the time the first impacts came.

As for the Vernon Square rocket, the massive radiation is likely from an exposed power supply. The model of it in the Museum of Technology mentions that it has a nuclear reactor as the power source.

Most of the allied countries were technically at war with China in the Korean war. Also known as the Forgotten War, which is probably why you're not familiar with it.

Americans, British and Australians fought North Koreans and Chinese there, this was a 1950's conflict, so it is safe to say the same mentality continued through to 2077.
Even after the Korean War had ended, Australians, British and New Zealand Special Forces spent a lot of time in South East Asia and in the Pacific area in general hunting down communist movements and stamping them out. A lot of Anti-Guerrilla operations were carried out by the Australians in Malaysia and other pacific countries. Its why they were so effective against the Viet Cong in the Vietnam conflict.

So its not really a mystery.

I can see people wanting to get them as well as use them. Ghouls for one and the Super mutants of the west coast; they went East after fallout one. They are not affected by radiation, Roy Phillips I can see using them. There is a crazy Mad scientist ghoul tying to Nuke people with an orbital weapons orbital platform, she might want to have some. Mr. burke and Tenpenny. Raiders would take anything. The Enclave would take them, if you knew of fallout's past so would any branch of the Brotherhood. I am not sure If you know of other Fallout games. In Fallout two the Enclave had one they were willing to use it to self destruct. The Master in Fallout One had one as well. In Tactics they use a nuke to get into Vault Zero. Nukes have been used in past Fallouts, hell Burke asks people to use megatones bomb! people are out their willing to use Nukes. There is the mini Nuke weapon. I think I have made my point.

Seeing how even Enclave and BOSO AI get their collective @sses handed to them on Hard and Very Hard by the sentry bots, robobrains, and protectrons guarding the place, it's no wonder any of the other factions would've even made it in there to steal the nukes. How Dukov, Crowley, and a bunch of random mercs and Raiders made it into the tech bunker is still beyond me.

-Repeat yourself this is just a show, I should really just relax-
-Mystery Science Theatre 3000-

But they did make it. Where there is a will there is a Way. People have got into alot harder bases in past Fallout games. If you can kill them and get in why can't anyone else in the wastelands? Yes the Character we make always becomes a SUPER HUMAN with his/her skills but others can get in as well as kill any Robot. The makers of the game just never thought about that little scenario. They just thought it would be cool to have a building full of bombs.