After bullets shatter the rear window of the car that Martha is driving, the next interior shots of the car show an undamaged window.
The first time you see the close up of the Master saying 'Gasmask' he has it on upside down as the logo is above the tube.
When the Master stands up on the Valiant to announce his that he is "the Toclafane's Master", President Winters is visible in the background, only the Toclafane aren't there. In the next shot, they return, buzzing around Winters' head.
The papers on the table shift position during the session of the cabinet.
Winters is described as "President", but identifies himself to the Toclafane as "President Elect of the United States of America". The President Elect is the candidate who has won an election (in early November) but not yet taken office (on January 20); he has no authority yet. (Russell T. Davies has stated that he used the term President-Elect without realizing what it actually meant, and that Winters is meant to be the full President of the United States.)
Right after the Doctor, Martha, and Jack time jump back to present-day England, Jack walks down the street with his handgun openly visible on his hip. Considering England's strict gun laws, it's unlikely someone wouldn't have called the police and reported him. However, this issue is frequently ignored in the New Whoniverse.
When the Master runs to the window to watch the spheres descend, he presses against the wall, which visibly bows out.
When the police open fire on Martha's car, you can see sparks where the bullets from their guns bounce off her windscreen. At that close range, just one of those bullets could go through a brick, and despite the back window of the car being shattered, there are no bullet holes in the sides of the car, nor have the side windows been shattered.
The Doctor says The Master is Prime Minister of Great Britain. Strictly speaking, Great Britain is a landmass not a country; what he should say is that The Master is Prime Minister of the United Kingdom.
It's stated that by the time of the events of Utopia (2007), all the stars had burnt out. When the rift opens to the end of the universe to allow the Toclafane to come to the present day, stars are visible on the other side of the rift.
(Around 38 minutes) the newsreader says "It's 3am in the morning on the Eastern seeboard." AM means in the morning so she should have said either "It's 3am on the Eastern seeboard" or "It's 3 o'clock in the morning on the Eastern seeboard."