I could not find out about a ban in Austria FWIW.
So, one of the first things that's important in terms of context for this movie is that in the aftermath of the liberation from Nazism and the founding of the Federal Republic of Germany, German bureacracy and government was very sensitve towards any sort of cultural material that dealt with Nazism in way, shape or form. Ostensibly and also in a certain sense in actuality that had to do with a sort of fear of any kind of media that featured Nazis stoking the flames of Neo-Nazis anew in the young Federal Republic. This was partly influenced by such things as the Zind case when in the late 1950s a prominent case of public anti-Semitism and the public reaction to it also lead to over 700 attacks on Jewish institutions and Jews in Germany. On the other hand, the same kind of senisbility encased within the German law for censoring certain media that counts as "glorifying Nazism" was used to stifle certain media critically of the Germans, at least by the generation of former Nazis in charge of executing these provisions.
Examples next to the late date of the showing of Cahplin's movie are the movies Casablanca and Alfred Hitchcocks Notorious. Notorious as you might be aware deals with Nazi criminals having escaped to South America and smuggling Uranium. In the German version, which premired in 1951 (5 years after the original release) under the title White Poison the Nazis through dubbing became drug smugglers and the uranium cocaine. Only in 1969 were German audience able to see a German version of the film that was faithful to the original story. Similarly, when in the same year Casblance came to German cinemas, almost all references to Nazis were cut from the movie. Viktor Laslo through the power of dubbing became a Norwegian physicist who had discovered mysterious delta rays and Cpatian Renault a member of Interpol. Why the guests of Rick's bar sang the Marseillaise was never explained and so forth. Only several years later did German audiences see the original version dubbed properly in Germany.
This is important as context for what German audiences were able to see in the years after the war and how they reacted to it, including The Great Dictator. In fact, the first showings of the movie are documented from Berlin in 1946 – one by a cinema who seems to have gotten hold of a copy from someone in the American occupation, much to the chagrin of people in the German press. The novelist Alfred Andersch and several other people protested that the movie was shown in Germany saying it was much to early to show a movie that portrayed Hitler in a manner that was funny – even satirically – so shortly after the war. Several of the argued form the position of victims of Nazism: Andersch f.ex. was forcibly divorced from his wife, who was Jeiwsh under Nazi German laws and lost several of her relatives in the concentration camps. Andersch himself deserted from the Wehrmacht in 1943 and started his career as a writer writing in a publication for German POWs. He and several other people who saw the movie were not able to grasp the humor and took special offence with the concentration camp scene (for which Chaplin later apologized).
Around the same time, the Information Control Section of the American occupation in Berlin arragned for two surprise test showings of the movie in Berlin cinemas. People coming to see a movie called Miss Kitty were instead shown The Great Dictator, observed during the showing and afterwards asked about the movie by the officers of the ICS who tried to assess its potential for using it as a tool in the denazification process. Reactions apparently were a bit mixed. While the audience tended to laugh about the portrayal of Göring and Mussolini and some of the scenes involving Hitler, after a certain point laugfhter seems to have stopped and having been replaced by awkward silence. Again, the scene of the movie featuring the concentration camp seems to have been the main trigger for the switch in atmosphere.
About 350 questionnaires of the 500 distributed by ICS were handed in again and while 75% of viewers on the first evening and 84% of viewers on the second night said the movie was "excellent" or "good" and 62 resp 69 percent of viewers said the movie should be shown in Germany. At the same time, the complaints and criticisms included mainly the German kauderwelsch Chaplin is speaking – this was identified as extremely hurtful by German viewers that someone would make fun of their language, the language of Goethe and Schiller. This among other things lead to a couple of dry humored comments by the Americans but seems to affirm for them to not show the movie on a broader scale in Germany after this feedback. It was deemed that things were still too early after all and Germans were not yet ready for a satire of a Great Dictator they cheered only a short time before.
From the 1950s showings when the movie was seen in cinemas properly for the first time no sources about compareable reactions survive. Reviews were favorable albeit critical of what they identified as "shallow Americna slapstick humor" and its incompatability with the topic. (I mean, who doesn't remember all the 1958 comedic classics from Germany such as Heinz Erhardt's Immer die Radfahrer, which I am sure is not shallow at all...) However, no further controversy seems to have occured.
Anyways, the question remains just how representative the ICD questionnaires would have been for the German population at large but what can be said that between 1946 and 1958 Germany already ahd changed a lot and it makes sense that while in 1946 reactions were highly mixed, cultural snobbery was more of a problem in 1958 than actual controversy about the Nazis.
Two more things to close this out: It's very likely that Hitler never saw the movie despite all the rumors that he did and maybe even liked it – he requested it twice to be shown for him but apparently they never could get their hands on a print...). The only organized German audience that saw the movie before 1946 was most likely a unit of Wehrmacht troops in Yugoslavia in 1943 – Yugoslav Partisans had amanged to inflitrate one of the Wehrmacht's mobile cinemas and had exchanged one of their propaganda movies for print they had received as part of a British-US airdrop to support the Partisans. Reactions of the Wehrmacht troops ranged from roaring laugther to threatening to shoot the projectionist who had no idea what was going on.
Thank you for the comprehensive answer.
Is there any documentation from Austria?