In 1918, while held captive by "moderate" guards, Alexandra says Nicholas' mother is "safe in England". In real life, the Queen Mother arrived in the U.K. in 1919, the year after the Romanov family's deaths.
In the beginning it shows the whole family going into exile at the same time. This was not the case, Alexi was sick at the time when his parents went into exile so he and at least one of his sisters stayed behind.
In 1917, when Anastasia is on the Siberia-bound train talking to her brother, Alexi, she says Britain's Prince Edward (later King Edward VIII) is "exactly my age." Edward VIII was born in 1894; Anastasia was born in 1901.
At the end, the narrator says "The trial started in 1939 . . ." The trial actually started in 1938.
The caption that says "Berlin, 1923" should say "Berlin, 1920".
At the start, the caption reads "Tsar Nicholas' Palace, near St. Petersburg", technically, that palace is in, what's technically part (and a district) of the city.
On the Sibera-bound train in 1917, Alexi says it never snows in London. The city gets snow regularly.