In international relations, as with bicycles, squeaky wheels get noticed. In the former a squeaky wheel can be a small-middle country's policies designed to attract the reluctant attention of great or even Super powers. In that sense South Korea's KSS-3 submarines (SSBs), carrying ballistic missiles, highligt the possibility of South Korean nuclear warheads within the next 10 years.
This strategically destabilizing prospect worries the hell out of the great powers (China, Russia and Japan) that surround South Korea and worries the US. This South Korean squeaky wheel also tempts the Royal Kim Dynasty of "communist" North Korea to do something. North Korean actions have never been subtle. A North Korean nuclear strike to pre-empt a nuclear armed South Korea is the very nightmare squeaky wheel conveys.
South Korea may be bluffing when it talks of nuclear propelled submarines (SSBNs). An SSBN is a extremely expensive submarine platform that only makes economic and deterrent sense if it carries nuclear weapons.
I think, if it is a bluff, it is designed to keep US troop numbers high in South Korea, keep maintenance payments to keep the troops there reasonable and mainly to maintain US reassurances that it will defend South Korea with nuclear weapons. North Korea is steadily ticking the boxes to create a capability to fire missiles with thermonuclear (read H-Bomb) warheads so South Korea is getting increasingly edgy. If US reassurances are unconvincing then South Korean SSBs with nuclear warheads will become more likely.
Meanwhile South Kore may wish to convey to the US that South Korean SSBNs would be a serious drain on South Korean defense spending. Money only spent because South Korea is desperate for continued US attention and nuclear protection.