Nostradamus' medical teacher at the university goes from functioning perfectly well to dead from plague within seconds. No disease kills that quickly, not even the most virulent types of plague or fever. Although he had been concealing his illness.
There is a globe on the table that shows north and south America very accurately, while there is a common myth that many people believed the Earth was flat during this period it has been well documented throughout history that people had known for many hundreds of years that the world was round. At no point would a highly-educated, albeit religious, family believe the world was flat.
Nostradamus tells Catherine de Medici that each of her sons will become King of France but she will outlive them all. In reality, only three of Catherine's then five surviving sons - Francis II (1559-1560), Charles IX (1560-1574) and Henry III (1574-1589) - inherited the French throne. Furthermore, she only outlived the first two of these. Henry III outlived his mother by seven months.
Nostradamus draws a swastika on the wall, which supposedly nobody has seen before. However, the swastika (or its reverse) was not new since the Nazis misappropriated an ancient good luck symbol.
Nostradamus is showing his son how to use a divining rod to find underground water and says that there is water 4 meters below, but the metric system wasn't invented until 1799 (roughly 160 years later).
When Nostradamus is brought to the town's theatre to watch a performance, there is a shot of the musicians playing. The music is correctly of the period, and the costumes of the musicians too, but there is a metal flute with keys being played. The flutes of the period would have been wood or ivory - metal flutes with keys were not invented until the late 1800s.