To add to this post of mine, it should be noted that there is a theory that the CIA helped killing Park Chung-hee because he wanted South Korea to have nukes. There is also a theory that the CIA helped killing Luis Carrero Blanco partially because he wanted Spain to have nukes.
Of course, both are dubious conspiracy theories but I found them worth of mention because if they are true, then, there is no doubt that the USA and the CIA would have done everything to stop or destroy Francoist Spain's nuclear program.
Regardless, IMO, Francoist Spain getting nukes is something that the USA wouldn't allow, even in the Cold War, because they already have allies with nuclear weapons, France and the UK, that are far better and saner allies that aren't far-right dictatorships. It would also annoy the Western European countries because they hated Franco and his far-right dictatorship. Many Americans wouldn't like this, either. It could also become Soviet propaganda. On addition, even USA relations with Francoist Spain were far from perfect and would definitely become worse if it did this.