While Isobel Crawley is officiating the move into the Abbey she is holding a ledger in her hands. When she talks to Violet in the day room the scene flips between looking at the two women, and the ledger flips from open (Isobel), and closed (Violet).
When Lavinia asks how long Lady Rosamund had lived in her London house, Lady Rosamund says she and her late husband bought it shortly after they were married. This is in contrast to what was said in the first season, when Lord Grantham refers to Lady Rosamund's "new house in Eaton Square."
Chauffeur Branson was rejected for military service due to a heart murmur which he called Mitral Valve Prolapse. A murmur would have indeed kept him out of the military but the particular disease called Mitral Valve Prolapse wasn't discovered until 1966 by Dr. John Brereton Barlow.
Lord Grantham states that he'll put some strings at the War Office to make William Matthew's soldier-servant. The correct term would be Matthew's "batman". Lord Grantham also talks about back at "base". This was an American term that didn't enter British usage until WW2. The correct term would have been "camp", or "HQ".
As Mr Carson discusses Thomas with Grantham (while Grantham dresses for dinner), Grantham turns to have Carson help the Earl with his jacket. The standing mirror captures three film crew and their equipment as the camera pans.
In the scene where Mr Carson helps Lord Grantham finish dressing after Mr Lang leaves the room, as they move in front of a mirror you can see the filming crew in it.