I don't see the UN being very useful after this. When would this happen? Nixon derided the UN in his presidential campaign. A more powerful John Birch movement might be needed. If Robert Welch hadn't called Eisenhower a Communist the movement might have been able to move into the Republican mainstream slowly over time. Then later, some issues involving Israel and perhaps Taiwan could provide the necessary momentum for the Bircher-Republicans to gain the political support to leave the UN.
UN would be increasingly dominated by the Soviet Union and other nations would likely opt out, such as the aforementioned Israel and Taiwan, but also there would be increasing dropout movements in other countries, such as perhaps Britain. In the West the UN would begin to be seen as simply a playground of Communists and rogue dictators. New international frameworks could easily spring up, NATO combined with ANZUS and other alliances. The Non-Aligned Movement may grow in prominence as the Security Council (dominated by the Soviet Union, the PRC and eternal minority France) becomes seen as a modern Communist Internationale.