Mission Accomplished!! The second nuke I built didn't get stolen! Yay!
It's very simple to actually dispose of a nuclear weapon, but takes a while to get one!
You can build a nuke yourself, but there's a VERY high chance it will be stolen!! That happened to the first one I built, it was stolen almost right after I built it, so I couldn't dispose of it.
Either build the nuke and take note of exactly when it will be done and dispose of it right away, or play offline avoiding FOB missions as detailed below.
OR--try stealing someone else's nuke! However, I've tried multiple times, and the only people who have nukes are the ones with ridiculous FOB defenses--save yourself some time and resources and build one yourself.
The requirements to build the Nuke are 75000 Fuel, 50000 Minor Metal, and 750000 GMP, of ONLINE resources. Your Offline resources are generally 10% of your online resources, so you'll need over 80000 Fuel total, etc.
To access the Nuke build option, you need to open up the Mother Base menu, select Resources, then the far right option gives you the choice to build Nukes and Dispose of Nukes.
Nukes take 21 online hours to build.
There is a trick you can use to avoid having your nuke stolen however, if you haven't passed mission 21, or haven't gone online.
If you never accept the FOB mission that occurs after mission 21, by disconnecting your internet whenever it comes up(very annoying, but it works) the FOB trial mission is cancelled and you CANNOT be invaded by other players and have your stuff stolen!! You'll have to disconnect to avoid the FOB mission every time you go to your chopper's command center, the ACC, or else play offline(disconnected). To build the nuke though, you need to be online and you can go online through the menu's, and as long as you go offline/disconnect before returning to your ACC you won't have to do the stupid internet disconnect thing to avoid the stupid FOB mission.
I finally went online near the end of the campaign.. built a nuke, and had it promptly stolen by someone near the top of the ranking leaderboards. Don't suffer what happened to me!
I personally recommend going online simply because you can gain more GMP and resources early in the game from your FOB and it's quite a bit of fun, but be warned--if you're online with FOB's and you build a nuke, it WILL get stolen quickly unless you dispose of it!