(at around 22 mins) In the opening scene, due to the floor falling, Apocalypse and the mutant power transfer chamber are dropped for over 20 seconds; that fall time would mean they fell 1.2 km max. When Apocalypse is rejuvenated however the camera appears to show Apocalypse only a few meters below the top of the pyramid which itself appears to be extremely close to Earth's surface. However, when the sunlight strikes the cap, the light lines race down the ruins, reaching far below (hundreds, if not thousands of feet) below the cap to wake Apocalypse.
Magneto's beard changes length throughout the film, even within single scenes. When Apocalypse has kidnapped Charles and they're standing on the rock, he goes from having a quarter inch (6mm) beard to being clean shaven.
(at around 2 mins) The opening scene shows the sun raising behind the funerary boat and the shadows match in the long shot. The close-ups show the same shadows from the North (90 degree change in direction). A latter scene shows the sun shinning directly in the entrance so the change can't be explained by a delay between the long and close shots.
(at around 1h 40 mins) In the world destruction scenes, Magneto's helmet appears to be rather metallic blue, but when Apocalypse hands it to him, it is coppery, perhaps burnt orange. In shadow, burnt orange/copper does not turn dark blue.
Hank/Beast's eyes change from yellow to red several times during the movie, especially during the final fight.
(at around 59 mins) Auschwitz seems to be abandoned and deserted. In reality the camp was turned into a museum in 1947.
(At around minute 5:30) When the temple is falling and people start running, someone yells in Arabic to run. In 3600 BCE, people in Egypt did not speak Arabic but Coptic.
(at around 1h 16 mins) When Quicksilver is running through the mansion saving all the students, he stops briefly to drink a soda that was left suspended in the air by the student he had just moved outside to safety. It is clearly shown that he can move faster than the force of gravity so this would be impossible because he would require gravity to cause the soda to pour out of the can and into his mouth, but because he is moving so fast, gravity would not have time to act and the soda (as he would perceive it) would remain virtually stationary in the air. He would have to pause long enough for gravity to take effect but in doing so he would most certainly be caught in the explosion.
When Quicksilver rushes Hank McCoy out of the exploding mansion, Hank's eyes blink.
(at around 30 mins) Since X-Men: First Class (2011), many characters have barely aged despite the passage of 21 years. Some mutants have been confirmed in earlier movies to age slowly. While it is not clear why Moira MacTaggert should remain youthful, Charles remarks on this astounding fact, thereby acknowledging the problem, if not explaining it, within the story.
None of the plot-holes related to time lines on this page are relevant. X-Men: Days of Future Past (2014) led to a new time-line being written. Which would most likely lead to causal effects that people wouldn't expect. Advancements in satellite technology, bands and film release dates changing and if there has been a great deal of battleground activities changes in nuclear programs. The premise of the previous movie was to rewrite time. So pointing out that things don't match up to how time was, is not a goof. It's effective and consistent with all causality theories.
(at around 2h 10 mins) Although Quicksilver wears an Atari shirt that is late 1990s/early 2000s design, he has been previously and purposefully dressed by producers in clothing from a later time period (see X-Men: Days of Future Past (2014) and his Pink Floyd T-Shirt) in order to make him seem "ahead of his time"
All alleged anachronisms and factual mistakes can be explained by the fact that the story, like everything in the X-Men cinematic universe, explicitly takes place in an alternate reality (Earth-295) than ours (Earth-616) where all supposedly 'wrong' music, fashion, items,technology, events etc. are all already in the time-line. The parallel 'dimensions' and various transgressions of the same conflict are often even essential to the plot development in both the comics and the movies and sometimes 'errors' are intentionally placed to accentuate the differences between worlds.
The opening takes place over 1000 years before the building of the Great Pyramids at Giza and over 800 before the building of the first attempt at a pyramid, the stepped pyramid at Djoser. However, the anachronism is explained by the post-credit scene of X-Men: Days of Future Past (2014) which showed a younger Apocalypse single-handedly making pyramids with his mind.
When Apocalypse and Storm appear in Angel's hideout, he screams "Verpiss dich!" at them in German, which is the case for addressing a single person, when he should be saying "Verpisst euch!", the proper case when talking to a group of people. This is the most blatant mistake with all German characters. The others speak in a rather discernible English accent.
Around 1:14, Alex wreaks havoc on Cerebro, but his shirt has no damage whatsoever.
(at around 44 mins) When Apocalypse collects the information of the current world from TV, a picture of the Big Buddha in Hong Kong is shown. The construction of the Big Buddha started in 1990, and finished 1993.
(at around 43 mins) When Apocalypse collects the information of the current world from TV, an Israeli army 155 mm artillery howitzer is seen firing a shell. In 1983 there was a war between Israel and Lebanon, however, this specific model was not in service until the mid 90's, plus the concrete slab on the left reads "Israel 2006" (in Hebrew). The footage is probably from the 2006 2nd Lebanon war.
(at around 32 mins) During communism in Poland the law was upheld by Militia rather than police. The scene in question is set in 1983 whilst Poland was still communist.
(at around 30 mins) When Magneto is packing, the passports are shown. The cover depicts an eagle with a crown. In 1983, under communism, the Polish coat of arms was an eagle without a crown. The crown was reintroduced only in 1990.
Stryker's soldiers can be seen wearing the Advanced Combat Helmet (ACH), which is the current standard issue helmet for the US Army. However, the ACH was not fielded until 2003. During the time in which the film is set, US military personnel used the steel M1 combat helmet, which was in use until 1985 when it was replaced by the Kevlar PASGT (Personnel Armor System for Ground Troops) ballistic helmet.
(at around 1h 3 mins) Quicksilver's Ms. Pac-Man (1982) figure dies, but the sound effect is that of Pac-Man (1980) dying, not Ms. Pac Man.
(at around 10 mins) In Ostberlin, one of Lada Riva cars was seen with illuminated side running lamps, which were exclusive to the Canadian market. The Soviet or East German version would have none of these.
In the helicopter, Nightcrawler says he can't teleport through the electrified cage, and Jean says she can't influence the crew's minds through it. So the crew would have been able to see the 3 extra mutants.
In the helicopter, Nightcrawler says he can't teleport through the electrified cage, and Jean says she can't influence the crew's minds through it. But Cyclops' eye lasers certainly would have worked to cut the cage.
Numerous characters in the movie come from other countries and speak other languages (Magneto's family is Polish, Nightcrawler comes from Germany) and these characters speak in their ethnic tongues, which is accurate. Yet said characters occasionally speak perfect English with no explanation. It is possible they learned English in schools as a second language, but they would not speak it this flawlessly.