Psychologically, this would have a few effects.
Firstly, it would feed into the idea of Ending Bloodshed/Staving off further Bloodshed/Preventing mass bloodshed with the Atomic Bomb. That was the reasoning for the A-Bomb in World War 2. And parallels could be drawn between the Vietnamese Communists and the Imperial Japanese; a violent enemy, barbaric in behavior, who would not surrender, and would fight on the scale of millions to the last man if necessary. The concept that the "Oriental" doesn't put the same value on life, as Westmoreland said.
Secondly, it would allow powers to get out of conflicts and quagmires by using the Atomic bomb. The Soviets could use it in Afghanistan, for example. They won't risk it with another nuclear power, due to the possibility of such conflict erupting into MAD. But against powers without Atomic weapons, likely.
What you've done, essentially, is removed the veil. Atomic weapons will lose a great deal of their taboo. They will maintain it insofar as not using them against other atomic powers, (because they can still destroy the world, and people will still fear them for that) and keeping their usage limited and tactical so it doesn't go too far and expand into a wider conflict. But the taboo of their use on a limited scale is removed.
Their continued use could continually make it so that the powers were looser about their use, which could slowly push towards being more readily willing to use them in a major conflict over time.