The Great Old One was once rammed with a sea vessel, and it did in fact appear to cause him great distress and take him down a notch or two.
I submit that a nuke could conceivably destroy him.
Some may differ, citing other of HPL's works that tell us that the Great Old Ones aren't quite made of the same kind of normal matter-stuff that we earthlings are, and could somehow survive even an atomic bomb attack... and maybe (*shudder*) even LIKE it!
(most likely in Azathoth's case, seeing as he IS "the Seething Nuclear Chaos")

Well, I do recall at least one story in which a nuclear attack was used on Deep Ones on Devil's Reef with "some" success, I really don't know if a nuclear attack would do much good against one of the Old Ones. As Lightbringer said, they're not even made of the same stuff as us. My theory is that they're of a much more primitive make up, like micro organisms or nannites, and if you didn't manage to get every little bit of matter, they'd just regrow (eventually) and be mad as heck. That might be the problem. Maybe someone blasted them to particles and it'll take till the "stars are right" for them to regrow enough to get the kind of revenge they desire.

Yeah, because they're not material in the mundane sense, and, that part of each GOO we drunkenly convince ourselves is 'corporeal' is nothing more than a malleable puppet-shell.

Cthulhu directly inspired the innsvention of atomic fission (and later fusion). Guessing at why is a pastime for fools and monsters.