"District 13 makes our nukes and guns that our nation has. Including landmines. This nation wouldn't be a nation without the 13th district of Panem."
Katniss Everdeen, on District 13's creation and destruction.[src]

Not much is known about Panem's nuclear program. We can assume that Panem's nuclear arsenal was a large one, as they dedicated a whole district to it. Another thing we can assume about Panem's nuclear weapons is that since The Hunger Games trilogy is set in the future, their nuclear technology should be more advanced than ours. Also, The Capitol is known to have nukes as a form of deterrent against District 13. No other nation is mentioned in The Hunger Games saga, but if there are other nations in The Hunger Games universe, The Capitol's nuclear weapons could be used to stop outside threats to the tyrannical regime.  

"They're us [District 12], if we'd had nuclear weapons."

 - Gale Hawthorne replying to Katniss on District 13.