This is an American-biased question. The US aided the UN, not vice versa. The war was between North Korea and South Korea.
South Korea was aided by the UN which provided contingents from:
Australia
Belgium
Canada
Colombia
Ethiopia
France
Greece
Luxembourg
Netherlands
New Zealand
Philippines
Thailand
Turkey
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The nations that were involved in the Korean war were the US who aided south Korea and the soviet union, who aided north Korea.
Member countries of the UN supplied troops and war materials to aid South Korea.
The UN didn't fight in the Vietnam War; the UN was in Korea.
They were in total support of South korea
UN member states