First off, the Memorandum of Understanding Between The United States of America and The Union of Soviet Socialist Republics Regarding the Establishment of a Direct Communications Link (1963) never established a telephone line. The State Department did bring up having a "white telephone" connection back in 1961, but the source of the red telephone seems purely to be the imagination of pop culture. That didn't stop it from showing up in a campaign ad for Walter Mondale in 1984.
The concern was that with telephone, leaders would need to use rapid translation, which potentially could cause misunderstanding. The MOLINK or MOscow Link used teletype. The Pentagon had one teleprinter, the Kremlin had another, and messages could be sent between each other in several minutes. (The White House Communications Agency has a backup console, and the White House was linked up so the president could operate it without physically going to the Pentagon.)
They operated -- and still operate today -- continuously, with pairs on the US side consisting of a Presidential Communicator and a Presidential Translator in 8-hour shifts.
During the Six-Day War, the Soviets sent multiple messages; here is an official translation of one:
Dear Mr. President,
Having received information concerning the military clashes between Israel and the United Arab Republic, the Soviet Government is convinced that the duty of all great powers is to secure the immediate cessation of the military conflict.
The Soviet Government has acted and will act in this direction. We hope that the Government of the United States will also act in the same manner and will exert appropriate influence on the Government of Israel particularly since you have all opportunities of doing so. This is required in the highest interest of peace.
During the same conflict, when the Israelis attacked a US intelligence vessel (the USS Liberty) for unclear reasons, Johnson used the hotline to send a message letting the Soviets know they were not planning on entering the war.
It did get some non-crisis use; Johnson had updates sent during the Apollo 8 mission, and in 1985 (after a fax upgrade) Gorbachev sent a 15-page handwritten letter to Reagan. Here's a sample from the translation:
With regard to Afghanistan, one gets the impression that the U.S. side intentionally fails to notice the "open door" leading to a political settlement. Now there is even a working formula for such a settlement. It is important not to hinder the negotiations in progress, but to help them along. In that event a fair settlement will definitely be found.
As already mentioned, in the 80s it switched to fax. In 2008 it went to email. It's been possible since the 90s to establish a direct voice link (DVL), but that's not considered part of the hotline proper. There's also now a Government-to-Government Communication Link, Nuclear Risk Reduction Center (RSSC) connection, and Foreign Affairs Link, all used for different levels of concern. The RSSC notably can now be used as a cyberwarfare link (and was used by Obama eight days before the 2016 election).
Other inter-country hotlines have been established over the years, some of them explicitly nuclear:
although a special mention should be given to North Korea-South Korea, with 33 telephone lines, the first being established in 1971 for communication between their respective Red Crosses.
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Arms Control Association. (2020). Hotline Agreements.
Blacker, Coit (1984). International Arms Control: Issues and Agreements. Stanford University Press.
Bohn, Michael (2003) Nerve Center: Inside the White House Situation Room. Potomac Books, Inc.
Bohn, Michael (1 August 2013). Hot Line: Even without a Cold War, the Washington-Moscow link is still up. The Washington Post.
Egilsson, H. Þ. (2003). The Origins, Use and Development of Hot Line Diplomacy. Netherlands Institute of International Relations.
Nanz, Tobias (2010). Communication in Crisis. The "Red Phone" and the" Hotline" BEHEMOTH-A Journal on Civilisation 3.2: 71-83.
