If the Soviets managed to have a public atomic test, before the Americans make their public, the US military and public will feel so humiliated that they will use their own bombs, in real battle, at the very first opportunity. Congress and the military would be screaming for them to be used in Korea, as soon as the first US soldier gets shot there.I doubt they would be used in Korea for the same reasons as OTL, they would be unlikely to accomplish much and would just cause respect for this great wonder weapon to diminish.
Imagine what a nightmarish scenario that would be - a humiliated US blows Korea to smithereens in an attempt to one-up the Soviets and restore the confidence of the public, then the Soviets in turn feel humiliated because their NORK proxy got atomized, and then they look for an opportunity to get even again. They couldn't escalate the war in Korea (what's left of it) for risk of hitting US troops, but the next third-world country that so much as looks at them crossed, will get turned into an irradiated parking lot.
Unless sane politicians step in, and hold some kind of high-level conference. I suppose at that point the UN atomic energy bureau gets founded, and non-proliferation treaties are set up. It would still be one hell of a tense period, more so than in OTL.