You might need an earlier POD.
Maybe something happens that shows how terrible the Saudi military is--I'd sum up their performance at Khaljfi during the Gulf War as brave and spirited but rather inept, but the U.S. did the heavy lifting to keep the Iraqis from capitalizing on this--and the U.S. is harsher on the Saudis post-9/11.
(With no Desert Shield/Desert Storm, 9/11 is probably going to be butterflied away. Maybe no Desert Shield, but fighting breaks out anyway and the Iraqis are slapping the Saudis, Qataris, etc. around before the U.S. intervenes and crushes them per OTL. We still get lots of U.S. forces in the Land of the Two Mosques and things go from there.)
The Saudi government decides that nukes are needed as a deterrent if they can't rely on their own conventional military or U.S. assistance.
Of course, the Saudis could simply buy nukes or the infrastructure thereof from Pakistan/the nuclear scientist Khan and his cronies.