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The Torah is read Mondays, Thursdays, and Saturdays; but it is also read on Rosh Hodesh (the first day of the new month), which could come any day of the week. It is also read on festivals and fast days. However, the Torah readings for Mondays and Thursdays are previews of the coming Saturday's full reading. Rosh Hodesh, festivals and fast days have their own Torah readings. A knowledgeable Jew would be able to tell by the specific Torah section that was read whether it was a weekly Torah reading or a special one, the latter of which could occur on Tuesday. Source: E.g. Art Scroll Chumash; Hirtz Chumash; Kitzur Shulchan Aruch; and more.
It is quite odd that the cantor, at the beginning of the Torah reading would announce the day (or more likely, have the psalm for the day read). Depending upon tradition, the psalm for the day of the week is read either at the end of the services (common in Ashkenazi synagogues) or close to the beginning of services (more common in Sephardi and Chassidic congregations).
There are two readings of the "curses" (aka the "warnings" or the tochacha), one at Leviticus 25:18-53 (within the reading called "Behar"), and Deuteronomy 28:15-68 (within the reading of "Ki Tavo"), each of which are followed by a description of how Israel can avoid or pull itself out from the curses by returning to God. These verses, however, are only read on the Sabbath. In 1940, Behar would have been read on May 18th. Ki Tavo would have been read on Sept. 21, 1940, or Sept. 2, 1939 (assuming that the filming was done earlier than the film's release).
Having not heard the recording or seen the film, I cannot verify any of this. I do think it unlikely that the Torah reader would call out the day of the week before the Torah reading. And I would think it fitting, but unlikely, that the Germans would just happen to bring their cameras when the tochacha was read on the Sabbath. If the whole thing was staged, and the director just asked them to read the Torah, then the selection of that reading would have been perfect.