Both countries have the technical expertise and the tactical need if push comes to shove, while having plausible deniability.
The conventional wisdom (pardon the pun) is that North Korea does not yet have a warhead small enough to fit on a ballistic missile. They would need another jump in miniaturization to get one small enough to fit onto a smaller diameter SLBM.
The danger here is that the US is using it’s own history as a basis for creating a timeline for how long all this will take to come to fruition. Copycats never take as long as those who originate a given technology.
South Korean culd probably build a nuke but it will take longer I think, the US probably watches them closely enough where they do not have components on a shelf somewhere.