This is not a mistake as such, but in real life the DJ who introduces 'Jump' (where Hugh Grant dances through No. 10) is Jo Whiley, who has a Radio 1 show on weekdays from 10am till 1pm. At this point in the film, though, it is dark outside, so she is not broadcasting in her usual slot.
The short interchange between Colin and the caterer at Peter and Juliet's wedding was originally a scene that was cut from the final version of Four Weddings and a Funeral. Richard Curtis liked the dialogue so much that he injected it into Love Actually.
Billy Bob Thornton agreed to appear in the movie without having seen the script. On the day of shooting he told director Richard Curtis that he was scared of Benjamin Disraeli's facial hair, and as such wouldn't be able to do the scene in Ten Downing Street because there was a picture of him on the wall. Curtis told him to keep his back to the picture throughout the scene, which he does, although you'll notice he looks quite nervous.
Rowan Atkinson was originally meant to be an 'angel of love' - he was meant to help all the movie's characters find their love (e.g. he helps Sam by distracting the airport employee). He was also meant to walk away at the end and fade away, but the writers thought this too complicated and decided to keep him as a simple shop assistant. DVD Commentary.