The United Nations (abbreviated UN), was an international organization comprised of nearly all nation states on Earth in the 20th and 21st century.
In 1953, the UN held a meeting in London. (DS9: "Far Beyond the Stars")
The original organization was succeeded by the New United Nations sometime prior to 2036. (TNG: "Encounter at Farpoint")
An organization known as the UN still existed in the early 22nd century, before the formation of the United Earth government in 2150. Starships, such as the OCC-owned SS Mariposa operated under the NAR UN registry in 2123. (TNG: "Up The Long Ladder", "Attached")
It's possible that these later references are rather meant to actually refer to the "new" UN instead of a second institution of the "old" UN.