The technical difficulties of creating a nuclear shell are actually pritty darn huge. What we eventually saw of the NK nuclear device they detonated was it was a very big thing, certinally not a weapon but a large and bulky affair. You'd need to be able to make the weapon itself before being able to do it.
As ye said Paul the South/US would win air supremacy within a few hours, but they would not be able to knock out all the artillery positions due to the sheer volume of them and the need to support troops on the ground. I agree on your timeline too, 3 weeks to two months max.
I've always kind of assumed that their greatest underground tunnel project was to get one of their nukes either as close as possible to Seoul, or to the projected route of advance for South Korean/US armored advance.
Maybe both.