When David visits Natalie's house her hair changes between the wide shots and the close shots of her on the stairs. In the both shots her hair is in front of her left shoulder but some pieces are in front of her right shoulder in the close shots that are not there in the wide shots.
At the recording studio, Bill is shown in one shot singing directly into the mic. In the next shot, he is standing about a foot above the mic. Not only is this a continuity error, but he also shouldn't have been that far from the microphone to be recording properly.
When Sarah and Karl are first kissing in her bedroom, Karl strips to the waist but still has his trousers on. In the next shot, as they fall onto the bed with Sarah sitting astride him, his trousers are missing.
The scene where John is trying to kiss Judy goodnight, but is too shy, they are wearing the same clothes that they wear at the concert. The concert is their first date. They go to the concert after they had their first kiss. It should have been the other way around.
When Sarah and Karl are in Sarah's room, Sarah's alarm clock over her bedside table says that it is five to twelve. A bit later we can see the same alarm clock and it is ten to twelve. (Karl's watch has another time altogether, but this is not necessarily a goof.)
The bar in Milwaukee inexplicably contains a pennant for the Midland RockHounds, a minor-league baseball team in Texas; posters for the Chicago White Sox instead of the Milwaukee Brewers; and several promotional displays for Budweiser, which would be an unpopular choice in Milwaukee, the home of archrival Miller beer.
When the Prime Minister is going to find Natalie at the 'dodgy end of Wandsworth' his limo is viewed crossing Albert Bridge, but is actually driving away from Wandsworth in the direction of Chelsea. The Albert Bridge has three lanes, with two going in the direction of Chelsea, although now the central lane is blocked to reduce weight on the bridge during peak hour (changed 2007).
When Colin visits the Wisconsin girls' home, the house is shown from the outside. On the street, a parked car is seen on the side of the street, facing the wrong direction. In America, unlike Britain, a parked car must be facing the direction of traffic, and all traffic drives on the right-hand side of the road. (This is not a traffic rule that is commonly ignored, and in Milwaukee, a vehicle parked facing the wrong direction would be ticketed and towed.)
The real door to 10 Downing Street does not have a keyhole for a key and has its hinge on the other side (it opens on the right rather than on the left). It also has a big doorbell button setting on the right hand side.
The PM watches Billy Mack on an episode of Parkinson (1971) - it is shown in "proper" 16:9 widescreen (obvious from the amount of space on either side of the chairs). Parkinson has never been produced (to date in 2004) in widescreen, even after the BBC started aggressively producing more shows in 16:9 after the advent of digital TV in the UK.
After swapping "no" and "yes" in English and Russian, Colin responds to "nein" with "ja." The correct "yes" answer to "no" in German is "doch." But Colin would probably not know this, since most English speakers do not.
When Jamie revisits Aurelia, a Marseillaise cab can be seen at the airport, yet everyone in the city seems to understand Portuguese. This can be explained, however, by a large Portuguese community in Marseilles, who live in a common area and would be the ones who understood the language.
In the scene where Jamie comes home and then abruptly leaves, the two little girls say, "I hate Uncle Jamie!" If you watch the girl on the left, she mouths the line just as the girl on the right says it, and then, she says it out loud.