Jimmy makes a 1 quid bet at 10 to 1 odds on the tug o war contest. This was a lot of money in 1921, equal to $100 in today's US dollars. As soon as he makes the bet, he recruits a fat man to be the anchor position on his team. The man he made the bet with never would have allowed a ringer to be inserted into the game after the bet was made. The odds would be completely changed.
When the family is at breakfast before leaving, all the make-up that Michelle as Mary was wearing in the preceding scene in her bedroom is gone, without her seen removing any of it.
The staff visit a fair in September 1921. At the fair, there is a swing ride. This ride actually debuted in Germany in 1972. A portable version of the ride debuted in 1975.
When Mary, Matthew and anna are in Mary and matthew's bedroom and Mary turns from the full-length mirror, a close look at Michelle's face shows she is wearing blush, eye-liner, mascara and eye-shadow. An upper-class woman in the 1920's would not have worn all that make-up.
When the family goes to Scotland for a holiday and the staff is downstairs, the butler of the family they are visiting refers to Mrs. Bates as Mrs. Crawley and and the housekeeper refers to Mrs O'Brien as Mrs. Grantham. It is traditional in English house parties for the visiting staff to be referred to by their employers' names. Therefore Mrs. Bates as Lady Mary Crawley's maid is referred to as Mrs. Crawley, etc.
When Matthew takes their son from Mary's arms, a fold in her nightgown is caught. When it settles back, Mary is shown to have a flat stomach. No woman has such a flat stomach that soon after giving birth.
"Gippy tummy" is used. While "Gippy" to mean "Egyptian" existed, "Gippy tummy" dates to WWII.
In learning to run the estate, Branson says that he's "on a learning curve". This phrase did not exist in the 1920s; it didn't come into use until the 1930s to describe productivity in the aircraft manufacturing industry.
When showing Matthew getting into the accident it shows the trees along the road he was driving on full and green even though it it set sometime in September/October.
When Mary says: "You'd better go down, they'll be here in a minute." to Anna, her words do not match up with her mouth movements.
Upon meeting Michael Gregson, Lady Grantham complements him saying she is an avid reader of The Sketch since Lady Edith recently started writing for the publication. However in the first episode of the first season Lady Grantham reads The Sketch in bed in the aftermath of the sinking of the Titanic, having had it freshly ironed by a footman.