When Mr. Grey and the Duchess meet by the lake (where they give their first kiss), the Duchess' hairdo is complimented with a rose above her right ear, which, according to the camera angle, is covered/not covered by her hood.
During the scene where Georgiana introduces her daughters to Bess a couple walking a black and white dog walk past. They are then seen further along the river before disappearing from the shot. They then pass in front of the camera again at the end of the scene.
Georgiana, Duchess of Devonshire was born in 1757. Charles Grey was seven years younger, born in 1764. According to the subtitle, the scene depicting a wager among the young ladies over a footrace between Charles Grey and other young men was held in 1774. Georgiana was correctly seventeen at the time, but Charles Grey was ten. He would have been a boy, not the young man about to attend Cambridge portrayed in the film.
The scene in which Georgiana goes into labor to deliver her first baby, begins at night on a snowy evening. This would have been her first baby girl that was later named little G, for Georgiana. However, the real Little G, was born on July 29th, far from what would have been a snowy evening.
Althorp Estate, Georgiana's childhood home (shown in the exterior shots of the foot race at the beginning of the film), was originally a red brick building. Red brick went out of style in the 1780's, at which point the entire building was altered with mathematical tile nailed onto the original red brick to give it its current gray exterior. At the time of Georgiana's marriage to the Duke in 1774, the building would still have been red brick as the architectural work started in 1788. The current Earl Spencer (Princess Diana's brother) had the mathematical tiles redone for a cool $8-9 million as many of the original nails used had rusted away, at which point the original red brick used for Althorp was also ruined.
In the last scene with the kids running around the fountain. Fountains have been created for hundreds of years running on gravity, not electricity. For example, the Trevi fountain in Rome was completed in 1762, and the Emperor Fountain at Chatsworth was completed by Joseph Paxton in 1844.
In the scene with the little girls in the carriage, when they print on the screen the names of Harryo, Little G, and Charlotte, they mix up Harryo and Little G's names. Harryo is the youngest, with the dark hair, but they put the name "Little G" in the front of this child and put "Harryo" in front of the blond older child. In a subsequent scene, Harryo, the youngest with the dark hair, falls and scrapes her knee. Georgiana specifically calls her by name, Harryo. In fact, Little G was older than Harryo. So the scene with the names is incorrect, while the scene with the scraped knee is correct.