Adapted from an older answer:

Knowledge about Nazi atrocities was widespread in Allied countries, both in the public and in the governments.

When the Einsatzgruppen in the Soviet Union began mass executing Jews in the wake of the German invasion, it took the Allied governments but one month to intercept, de-crypt and transcribe them. By mid-July 1941, the British government was fully aware of the activity of the Einsatzgruppen, which they reported to Berlin via HF Radio and which the British soon very able to decipher. The Einsatzgruppen Situational Reports USSR as they are known are incredibly frank and open about Nazi anti-Jewish policy in both the Reich and the Soviet Union. Take for example the first report the British intercepted, Report No. 101 from October where concerning the massacre at Babi Jar, the Einsatzgruppen report:

Sonderkommando 4a in collaboration with Einsatzgruppe HQ and two Kommandos of police regiment South, executed 33,771 Jews in Kiev on September 29 and 30, 1941.

This is but one of these reports where they openly state that they are killing thousands and hundred thousands of Jews.

Similarly, the Western Allies were aware of the Operation Reinhard Death Camps when they started operating in 1942. The best example for this is the so-called Höfle Telegram, a precise statistic on how many Jews had been killed in these camps until December 31, 1942 which was encrypted on an Enigma Machine and thus deciphered and intercepted by the British. While British code breakers apparently missed its relevance, the Polish government in Exile and their Commission for the Crimes against the Polish Nation did indeed pick it up in its significance and duly reported to London about it.

Especially the Polish government in Exile did a lot of further work to spread the knowledge of these crimes, going so far as to have members of the Armia Krajowa smuggle out reports from the Auschwitz Concentration Camp. And while it is right that the London government largely dismissed these reports that was not the result of disbelief necessarily but rather of the official position that it was important to concentrate on winning the war militarily in order to end this rather than focus on this particular plight too much.

That the Allied governments were aware of what was going on is also revealed by their policy surrounding the German occupation of Hungary in 1944. There for example a Swedish diplomat, Raoul Wallenberg, handed out 30.000 Swedish passports to Jews in Hungary in order to save them from deportation with the very precise knowledge that they were going to be killed. Similarly, the British government several times initiated negotiations with some of the Axis countries to save Jewish children by transporting them to Switzerland and Palestine.

As for the public in the US and the UK, they too were aware if not of the fine details, that Germany was killing Jews in large numbers. The New York Times e.g. published an article in 2001 admitting to its own failure to report more prominently on the Holocaust. They wrote:

Why, then, were the terrifying tales almost hidden in the back pages? Like most -- though not all -- American media, and most of official Washington, The Times drowned its reports about the fate of Jews in the flood of wartime news. Its neglect was far from unique and its reach was not then fully national, but as the premier American source of wartime news, it surely influenced the judgment of other news purveyors.

While a few publications -- newspapers like The Post (then liberal) and PM in New York and magazines like The Nation and The New Republic -- showed more conspicuous concern, The Times's coverage generally took the view that the atrocities inflicted upon Europe's Jews, while horrific, were not significantly different from those visited upon tens of millions of other war victims, nor more noteworthy.

(...)

Only once did The Times devote its lead editorial to the subject. That was on Dec. 2, 1942, after the State Department had unofficially confirmed to leading rabbis that two million Jews had already been slain and that five million more were indeed ''in danger of extermination.'' Even that editorial, however, retreated quickly from any show of special concern. Insisting in its title that Jews were merely ''The First to Suffer,'' it said the same fate awaited ''people of other faiths and of many races,'' including ''our own 'mongrel' nation'' and even Hitler's allies in Japan if he were to win the war.

Following the less than enthusiastic coverage of this topic, on March 9,1943, screenwriter and Zionist Ben Hecht staged the play We Will Never Die in Madison Square Garden in front of 40.000 people in order to raise awareness of the plight of European Jews and then further traveled around the US with it, even winning over Frank Sinatra to participate.

In Britain too – though complicated by British media laws – the public was aware of what was going on if they chose to read the newspapers. The Daily Telegraph reported in 1942 about traveling gas chambers, which given that the Einsatzgruppen did indeed use gas vans is surprisingly accurate. Simon Leader's 2004 PhD Thesis on the British regional press and the Holocaust (pdf warning) shows that

[British] newspapers were fully aware of the Nazis’ intention to murder all Jews under their control by December 1942. They all reported the events that came to be understood as the Holocaust, (some in extraordinary detail) but the Manchester Guardian stood apart because of the consistency of its coverage

Concerning the expressions of surprise often cited from soldiers liberating camps when they came across them, for the soldiers who liberated these camps, the abstract knowledge of horrible things occurring was something definitely present if they were avid newspaper readers. Seeing it however, was something completely different. Even Eisenhower who definitely had heard about the intel collected was still completely shocked by what he saw because, once again, knowing something exist in the abstract is something different than actually beholding it.

Something similar can be applied to Casablanca and other movies e.g. Ernst Lubitsch's To be or not to be (also, Casblance was based on an unproduced 1940 play Everybody comes to Rick that was written before the Holocaust started): In the sense that Allied publics were aware to an extent of Nazi atrocities but the actual concrete pictures and extent of it came as a major surprise once camps were actually liberated.

Sources (aside those mentioned):

• Robert J. Hanyok: Eavesdropping on Hell: Historical Guide to Western Communications Intelligence and the Holocaust, 1939-1945, 2004.

• Witte, Peter; Tyas, Stephen (Winter 2001). "A New Document on the Deportation and Murder of Jews during "Einsatz Reinhardt" 1942". Holocaust and Genocide Studies. Oxford University Press. 15 (3).

• Jan-Erik Schulte: London war informiert. KZ-Expansion und Judenverfolgung. Entschlüsselte KZ-Stärkemeldungen vom Januar 1942 bis zum Januar 1943 in den britischen National Archives in Kew (Fundstück), in: Rüdiger Hachtmann u. Winfried Süß (Hrsg.), Hitlers Kommissare. Sondergewalten in der nationalsozialistischen Diktatur, Göttingen 2006 (Beiträge zur Geschichte des Nationalsozialismus, 22), S. 207-227.

Follow-up question: In the film, more than nothing is made about the character Victor Laslow's escape from a concentration camp. Do we know if the filmmakers understood the full gravity of what that would have entailed?

I'd like to chip in with my 2 cents here by saying that discussion about concentration camps was common in the Czech discourse prior to 1945, and even prior to WWII itself.

For example, here is a snippet of a public print conversation between Rudolf Medek (legionnaire and writer) and Karel Čapek (journalist and writer) from 1934, where they discuss (among other things) German concentration camps as an institution of the Third Reich.

It appears that the existence of concentration camps in Germany was taken as a matter of public record.

Furthermore, it appears to have been publicly known what the intention of the camps and other 'institutions' was. Here is a 1938 article calling for a solution to the Jewish question in Czechoslovakia, which says: "It is out of the question that Czechoslovakia should, in that sea of nations seeking purification and liberation from the Jewish minority, stand against all and risk its fate for the Jewish question."

The way I read it, it appears to have been generally known what was going on. The fact that nobody did anything to stop it is a supplementary question, which I suppose comes down to antisemitism outside of Germany (which, after all, was incredibly strong) and fear of antagonising Hitler.