Joseph Mohr wrote the poem of "Stille Nacht, heilige Nacht" (english: "Silent Night") already in the year 1816, before he went to Oberndorf. Franz Xaver Gruber just added the sound to this poem. The movie clearly shows Joseph Mohr writing the text inspired by a recent experience he had just before Christmas.
The inspiration for the lyrics of "Silent Night" was, like in the movie, the birth of a son, but it happened in 1816, two years before the song got the melody by Franz Gruber.
Joseph Mohr was around 20 years old when he started his job as a minister in Oberndorf, while in the movie he is portrayed by a 38 year old Tobias Moretti.
Johann Burgschwaiger shows his wife blueprints of a railway steam engine and the train furniture. The story is set in 1818 while the first railway connection, the Liverpool and Manchester Railway, first came in mind in 1823. So it is very unlikely that Burgschwaiger would have those blueprints in handy.
The scenery of the whole movie is set in a mountainous area like the Alpes. Oberndorf near Salzburg is located north of Salzburg in an area of a few hills, but no mountains. It would take a walk of more than one day to reach a mountainous area as shown in the movie.