Mickey identifies the HMS Taurean as a Trafalgar class submarine. The weapons selection screen shows that the sub is carrying Polaris and Trident ballistic missiles. The Trafalgar class is an attack submarine, designed to hunt and kill other submarines. It would not be carrying ballistic missiles as part of its weapons load.
The weapons override screen of the HMS Taurean shows 8 tubes for weapons. The Trafalgar class has only 5 torpedo tubes.
Mickey fires a UGM-84 missile at 10 Downing Street at the Doctor's instructions. The UGM-84 Harpoon is an anti-ship missile and not a land-attack missile. A variant of the Harpoon has been designed as a Land Attack Missile (SLAM) but is not in service with the Royal Navy.
Every nation that currently has nuclear weapons has complete autonomy over when/where they get used. There is no protocol that requires a vote from the United Nations in order to launch them.
Obvious disparity between CGI Slitheen creatures and the live action during chase scenes - the monsters moving at far faster speed than their quarry, yet repeatedly repositioned a considerable distance behind again in subsequent shots with the full-size (and different-colour) monster costumes.
When the boy is cleaning off the graffiti on the TARDIS, he opens the door and you can clearly see the glass windows from the inside.
When the Slitheen at Mickey's flat unzips his head, the blue light shining from his forehead fills the frame and seems to cast a shadow BEHIND him. Unless he was standing in front of a mirror, this would simply not be possible.
At one point Harriet Jones talks about The British Isles as if it was a nation; it isn't, it's a geographical term.
When Harriet Jones brings up that the Slitheen 'fart' doesn't smell like a fart, Rose assists her by saying "bad breath". Rose was never in the same room with a Slitheen when there was a 'gas exchange'.