In our world, nations known as the nuclear powers (sometimes called "Nuclear Weapon States"), are those countries which are known to openly possess stockpiles of tactical and strategic nuclear weapons.
As all of the Fallout games take place in an alternate universe that diverged from our own after the end of World War II and as early as 1947, it is unknown how many and which states of the world in that universe had access to nuclear weaponry or if any nuclear treaties were ever signed. Given the available information on pre-Great War countries in the Fallout universe, it is fairly obvious that both the United States and China possessed them, though they were likely nuclear bombs and nuclear-tipped ballistic missiles that possessed much lower explosive yields and much higher amounts of radioactive fallout than the weapons currently in those nations' arsenals in our own reality. The Soviet Union, which survived long into the twenty-first century in the Fallout universe and did not collapse in 1991 as in our world, is implied to also have possessed nuclear weapons, and during the Resource Wars that began in 2052, the European Commonwealth and the oil-rich nations of the Middle East both suffered from a limited but mutual nuclear exchange. It is known that terrorists have also been able to gain access to nuclear weapons, as per the terrorist strike that destroyed Tel Aviv, Israel with a nuclear weapon in December 2053.