I can only speak for Germany with certainty, but I think it is not so different from other places.

The word holocaust was usually only known to people with humanistic education, i.e. people who knew ancient languages, especially ancient greek and had some interest in religion and religious history, since 'holocaust' is a milennia-old word for 'burnt sacrifice'. It had been in use already in connection with the Armenian holocaust and even with the persecution of the jews by the Nazis, but mostly among jews or very highly educated people.

People in Germany after the war talked, if at all, about the "Judenverfolgung", the 'persecution of the jews' or the "Völkermord an den Juden", the 'Genocide against the jews'. There was knowledge about the genocide, especially after the Eichmann Trial 1961 and the Frankfurt Auschwitz Trials 1963 - 68, but that wasn't that well received, people felt mostly uneasy about what we call 'holocaust' or 'shoa' today. If it was discussed it was usually somewhat clinical and abstract (the media) or with anger and lots of denial. You usually heard 2 statements by the people: "Irgendwann muß doch auch mal gut sein damit" (hard to translate, something like ' We have already talked enough about it'), the so called "Schlußstrich-Debatte" ('point double bar-debate') and "We have always been against Hitler, it was those other social groups that just had to elect Hitler. And once he was in power, what choice did we have."

And to the younger people, once they started to ask questions and to question the authorities - parents, the churches, the military, the government, the police, whatever - that also had been spoiled by the Nazi past: 'You haven't been there, you don't know how it was'. There was a lot of bad blood and resentment on all sides of the discussions, if you could even call it that. (I'm of course not talking about 100% of germans but about a very significant part of them. And of course you can still hear those 2 statements, but just not nearly as frequently anymore)

Comes January 1979, comes the TV mini-series 'Holocaust'. (There have been bomb attacks on 2 TV towers by Neo-Nazis)

Suddenly the victims had faces and the atrocities became concrete. People who prevously hadn't cared that much suddenly cared, people who had been nothing but stonewalling became tottering. The TV broadcasted panel discussions after every episode, with viewers calling in. And people were discussing the episodes at work, in schools etc, it was one of the most effective moments in german TV history. It did by far not change everything, but it triggered a lot.

And that's how people started to use the term 'Holocaust'. The documentary 'Shoa' from 1985 did btw the same for the word 'Shoa', that is in use in Israel since the 1950's, but it didn't have the same impact as 'Holocaust', the documentary wasn't just that accessible.

I just had a talk about this with scholar Dan Michman (former professor at Bar Ilan University and currently of Yad Vashem) and this is what we discussed. The term "holocaust" began to be used to describe disasters (both natural and man-made) even before the 20th century, and was often found in descriptions of massacres and natural disasters alike in the early 20th century. From 1933 on it was used, particularly in French to describe the Nazi treatment of Jews. Most Eastern European Jews used various Yiddish words during and after the period to describe their experiences, particularly Churbn, which means destruction, and in Hebrew the most frequently used term became Shoah. The term Holocaust became the primarily used term in English by the late 1950s when it was used with an uppercase h (the Holocaust) in the introduction to Eli Wiesel's "Night" and in relation to the 1961 trial of Adolf Eichmann. The term became even more used after the mini-series "Holocaust" in 1978 in the US and 1979 in Germany. From then on the term Holocaust and Holocaust survivor became the generally accepted term in most circumstances, though Shoah is still preferred in Hebrew and French. Some Jewish scholars prefer Shoah or Churbn because of the original meaning of the term "holocaust" which is a Greek word for burnt offering, it is the Greek translation of the Hebrew word "olah" which generally has a more positive, or at least holy meaning than most feel should be associated with the actions of Hitler. Associating a religious, purifying experience is seen by many scholars as problematic because it makes the act holy, which is wholly unwarranted when it comes to genocide. However since the meaning has fairly radically shifted towards a description of the worst horrors imaginable, and because it is the most widely understood term, it will most likely continue to be the preferred term for most scholars.