The London which Cora and Luke walk through is pristine, with bright white buildings, crystal clear air, and spotless streets. In reality, 1890s London was a filthy city with smog choked air and horse droppings and dirt in the streets.
When Dr. Garrett performs open-heart surgery, he injects adrenaline in an attempt to revive the patient from cardiac arrest. Adrenaline would not have been available to Dr. Garrett or any other progressive surgeon in 1893. Adrenaline was isolated and identified until 1895, and was not available commercially until 1901.
Cora Seaborne stated that eels swim upstream from the Sargasso Sea and come to England to breed. When the show takes place in the late 19th century, the life cycle of the eel was a mystery. (It wasn't until the late 19th century that leptocephali were even recognized as eel larvae.) In 1904, Johannes Schmidt tried to track the origin of leptocephali, and it wasn't until 1922 that he discovered small ones in the Sargasso Sea and theorized that this is where they originate.
You can clearly hear the Carousel Waltz on a music box at approx 29:10 but this was not written by Rogers and Hammerstein until 1945 for the musical Carousel.