Now, making the Vatican a nuclear nation, or even the only nation (but not first), with nukes is easy enough. Just have treaties limiting the use of nuclear weapons and have them transfered to Vatican control (i.e., in place of say, UN control). Making them the FIRST with nukes is very close to impossible, but here goes:
A series of extremely large and deadly wars destroy much of Europe late in the 19th and early in the 20th century. Technology advances a bit faster than OTL, with OTL 1917 tanks by 1899 or so, and bombing raids like Dresden occur by 1920. Tired of war, the states of Europe with a Catholic majority (mainly France, Italy, and Spain) join in a 'Catholic European Union,' indirectly controlled by the Vatican but directly controlled by their own kings, queens, parliaments, etc. Much like the Holy Roman Empire less Germany plus Spain, around 1928.
The other nations of Europe do not join any unions, and the entire continent has been destroyed. America has not entered any wars.
The British Empire and the Catholic Union Empire are the most powerful remaining political units. Wary of each other, they enter an arms race. By about 1958, the Catholic Union Embassy representatives in London announce that they have a "bomb of unimaginable power" ready to test. In a demonstration for their British counterparts, Catholic Union scientists detonate a nuclear weapon in the sands of Algeria.
Sound good? Now the Vatican (sort of...) has the first nuclear weapons.