Stamets says Starbase 1 is 100 AU from Earth, yet when Discovery arrives, there is a well-lit planet seen behind the Starbase. 100 AU (9.3 billion miles) denotes a location beyond the orbit of Pluto, but within the sphere of the Oort Cloud. There are no planets there, and if there were, they would not be so well-lit by the light of our sun.
When they reach the moon in the Veda system, Lt. Detmer locks the Discovery into a geostationary orbit, which would mean that the Discovery would always be over the same point on the moon's surface. The display clearly shows that this isn't the case.
Admiral Cornwell orders, "All evidence of your recent journey will be classified and destroyed." There's no reason to classify evidence if you're only going to immediately destroy it.
When Burnham brings up the holographic image of the Klingon planet of Kronos it has what seems to be an image of Earth as shown with what looks like South America being displayed.
When Georgiou walks onto the bridge, the admiral announces that although she had been thought dead, she had recently been found alive in a Klingon prison. The bridge crew beams with pleasure, with apparently all of them forgetting they had just been to a mirror universe with versions of all of them and none of them even seeming to suspect that it is the mirror version of Georgiou.
Admiral Cornwell says that Discovery was destroyed nine months earlier, and that she saw the debris. The crew explains that what she saw was the ISS Discovery of the Terran Empire, which had switched places with them. Even if none of the debris had Terran markings visible (including the name of the ship on its hull, with the "ISS" replacing the "USS"), the Discovery was able to determine that were in an alternate universe by almost immediately identifying that nearby Klingon debris had a different quantum signature from theirs. Surely any Starfleet vessel that came across or investigated the wreckage of the ISS Discovery would have been able to make the same determination.
When Cornwell asks Stamets to jump them to Starbase 1, he bemoans that it's 1 light year from their current position. A trip of that distance would be very short for a warp-capable Starfleet vessel. Even at the warp scale of the original series era, it would take just over an hour.
Sarek mispronounces Georgiou's name as "Georgia."