“...under the United Nations Command...”
There was no UN military command. The UN command was all political. The military command was all USA. I was there 65 years ago. It was not called a war, but it was a hard fought, deadly war. We took huge numbers of casualties right up until the day of the cease fire. The two primary countries engaged were the USA and South Korea. And believe me, the South Koreans were, and are great soldiers. There were others but they were in much smaller numbers, the sum total of all the others was regimental or brigade size. There were Brits, Aussies, Canadians, French, Turks and some others. An interesting feature, I thought, was that the South Korean government placed their entire military force under direct control of the United States commanders.
It still is, to this day. The United Nations/USFK/8th Army/CFC commander is, and always has been, an American Army general, since General of the Army McArthur held that position.