Other mistake: When Warwick Davis shows Joe the form he's tricked his father into signing to allow him to transfer schools, the form is shown full screen. The "home" phone number is the same as that used in the school's letterhead, and there is text that says"insert name of fictitious school here." Also in this scene the name of the headmaster and phone number is visible, but none of those details match what is on the school's advertising board a couple of scenes later when Joe starts at his new school.
Other mistake: Daniel Blake in the first minutes of the movie is at the hospital taking an echocardiogram. On screen you can see the date of the test; 20/10/2015, so he took it on the 20th of October. The day goes on with Daniel visiting his coworkers, then he is at home, with BBC Radio Four saying it's twelve to 1:00 and broadcasting the first shipping forecast of the day for "Tuesday the 20th." (00:04:40 - 00:06:50)
Other mistake: The tow truck driver says he has to repair the "split axle" before they can proceed. It wasn't clear what kind of truck he arrived in but the car could easily have been towed on the rear wheels on a wrecker and winched onto a flatbed.
Other mistake: During the scene when Jack's wife is ordering at McDonald's, the little girls are mouthing the lines as the others are saying them.