When the Doctor visits Rose at the end of the episode, she tells him the date is "2005, January 1st". Rose then runs into a building where on the glass by the doorway is a sticker stating "NO SMOKING. It is against the law to smoke on these premises." This type of sticker only appeared after the smoking ban in England came into force on 1st July 2007, two and a half years after it appears in Rose's building.
When Wilf is gazing out of the window of the spaceship at Earth, for the first time, the Doctor runs past and around the corner before coming back around after a moment to speak to him. However, it is obvious that David Tennant simply stops around the corner and pauses, before turning back. (Notice his feet are still visible, and the unnatural way he strides out.)