Audio problem: At the very end of the movie when Tony Blair and Her Majesty are walking in the garden with the dogs, the audio of Blair's voice does not match his mouth. At some points his mouth is not even moving at all, yet we still hear the audio.
Audio problem: In the very final scene where the Scottish pipe band is playing the tune "Amazing Grace", it is apparent that the music is overdubbed and professional musicians were not used for the visuals. When the pipers are slow-marching, the marching is shoddy with different people stepping at different times ("experts" would be very precise). As the camera pulls back past the drummers, it is clear that some of them are beating their drums at a very different time to the sound track.
Audio problem: When Millie comes to see the grieving Beatrix, she presses her face to the door and pleads with her to come out. If you watch the mouth movements of her shadow, they do not match what you hear her say.