Although this episode begins at a point after the main events have taken place, there's still an error with the Queen's hairstyle, of which much is made - both in the styling scene itself and Prince Philip's reaction to it. It's cut into the shorter, curlier style Her Majesty has favored to this day, but as the episode progresses, the hairstyle on actress Claire Foy reverts to the original, longer style.
When the Queen goes hunting, we see her POV through the rifle scope, which is moving freely with the stag's movements. All other shots show the Queen aiming the rifle completely still.
TV sets did not turn on instantly in the 1950's, that did not happen until tubes were replaced with solid state electronics in the 1970's.
The Jaguar Mk1 wasn't called the Mk1 at that time, just "2.4" or "3.4" depending on engine size. The Mk1 moniker came about after the Mk2 was introduced a couple of years later, being a continuation of what would then be called the Mk1 to distinguish the two generations.
When Queen Elizabeth shoots the stag, the stock of the rifle is under her arm, not at her shoulder. Shooting a rifle in this manner would likely result in injury to the shooter from the recoil; including being hit in the eye by the scope.
Characters are all sitting down to watch an episode of "Impact" which we are told several times is airing at 9:00 PM. However, the sun is brightly gleaming through several windows and lighting up the rooms of the palace.
As Michael Adeane is reading the prepared draft for the Queen's speech to be given at the Jaguar factory, he adds (in pencil) "as a whole" to a paragraph where the words are already typed down.
The drummers in the pipe band are playing high tension snare drums with Kevlar heads. In the 1950s, drummers played rope tension drums with leather heads.
During the first television broadcast of the Christmas address, the Queen is shown sitting at a desk and crossing her right leg over her left leg at the knee. The Queen would only have crossed her legs at the ankle. This is actually a continuity error as well, the crossing of the leg occurs in a long shot. In the close up that immediately follows, she is crossing her legs again.
At the Jaguar factory the Queen is shown a Mark 1, often rendered as Mk 1. But the person showing her the car calls it an M.K.1.
In her Christmas broadcast, the Queen misquotes from "Pilgrim's Progress", saying "I repent me of all the trouble I have been at to arrive where I am." In the actual broadcast she said (quoting correctly) "I do not repent me....."