@10:29 minutes, Sarah's teacher exclaims: "Homework is due end of class tomorrow!" September 26th, 2003 was a Friday. "Tomorrow" would have been a Saturday.
At 46 minutes, when Ellie kicks the tray it lands on the doctor's right foot, in the next shot the tray is on her left foot.
(around 1 hour and 5 minutes) When Joel pushes Ellie against the wall, the position of his arms differs between shots.
The teacher tells the students that their homework is due at the end of the following day, but Uncle Tommy begs Joel to get him out of jail that night because it's Friday and he doesn't want to be stuck there all weekend. In 2003 you couldn't turn in homework online so it doesn't make sense for the teacher to ask for homework on a Saturday.
When Sarah gives Joel a glass of orange juice to go with their breakfast, he takes a small sip then picks up his coffee cup again. He picks it up with his fingers gripping around the top of the cup. When the camera angle changes, he is holding the cup by its handle.
When Sarah goes to the Watchmaker in order to fix Joel's watch. He states that it's an easy fix, all he needs to do is "replace a spring", in the previous scene you can clearly observe the watch is a Quartz watch. Those do not have springs, they operate on batteries. The watch maker should have probably checked the battery first and just replaced it. He could have said: "It's an easy fix, I just need to replace the battery" . That would probably cost the $20 he charged Sarah.
When Sarah gives Joel his watch as a birthday gift. Joel says that the watch is not working while taking the watch jokingly to his ear and saying: "Can't hear anything". During a previous scene, when the Watchmaker opened the watch it was visibly a quartz watch. Quartz watches contrary to Mechanical watches do have a ticking sound.
At 37m57s in, a female worker who is incinerating bodies takes a break. She lowers her bandana facial covering, and wipes a gloved hand under her nose. This unhealthful act negates any protections the bandana was affording. This is incorrect as the bandana was to filter the smell of the burning bodies, not to stop infection.
The folks to be hanged are said to have been sentenced to "death by execution." The sentence would have been death; execution is the carrying out of that sentence by some method that would have been specified. "Death by execution" is equivalent to saying "death by killing."
The song playing in the background of the Lone Star pawn shop is "I Can't Believe You're Back" by Jad Mhanna which was released in 2018. Not 2003.
The bookshelf in Nana's house contained several DVDs when Sarah was looking for a movie to watch with Joel. Although the prologue is set in 2003, there are copies of Murderball (2005), a remake of Fama (2009), and Los Boxtrolls (2014). Those films were released in 2005, 09 and 14 respectively.
One of Joel and Sarah's neighbors in 2003 has a Little Free Library posted in their front yard. The first Little Free Library wasn't actually built until 2009, by Todd Bol, as a tribute to his late mother. (Duplicate.)
During the 2003 segment, Sarah is seen using a PaperMate InkJoy® pen to take notes in class. The InkJoy® brand of pens wasn't released until 2012.
The sign for highway '183' is the wrong type. In Austin, Texas, US 183 would have a black-and-white sign instead of the red-over-blue interstate sign seen in the truck headlights. Another sign, for FM 58, refers to a road that is in Lufkin, Texas, not Austin.