The gaslights used in the movie are all naked flames. By the 1930s, gaslights used mantles, which gave off a lot more light for less gas and didn't smoke the glass covers. Also, gaslights were largely controlled by clockwork (still used in historic areas today in London and other European cities), so the leeries' main job would have been maintenance: mantle replacement and winding the clockwork rather than daily lighting/dousing.
In the London street scenes, several of the buses shown are too old for the 1930s setting. For instance, the buses are shown with outside staircases to the upper deck, whereas those designs (from post-WWI) had been superseded by then by designs with enclosed staircases.
When the children are going to the shop and don't have enough money, they mention buying things at 'half-off'. This is an American phrase that would not be used by a child in London who would say 'half price'.
When Mary Poppins lands with the kite, she is holding the kite back-to-front, with the struts facing forward. If one has ever made, or flown a kite, one will know that this impossible - a kite cannot fly like this.
In the original Mary Poppins, the kite is mended with newspaper along its right border. In the new movie, the kite's original yellow border is seen all the way around.
This is a sequel, not a remake, therefore it would not be expected for the kite to be exactly the same.
When Cousin Topsy performs a handstand during her dancing routine, her earrings do not dangle down to the ground. This shows the scene was filmed with her putting her hands on a surface above her head and subsequently being rotated 180 degrees in post-production. This is clearly intentional as the rest of the scene plays out with all of the characters standing on their heads and everything remains intact, i.e. the skirts of three female characters.
The goof items below may give away important plot points.
It's stated that many foreclosures equals big profits for the bank. This isn't true; many foreclosures means many bad loans, and many bad loans is fatal to a bank. Banks make their money by the interest they charge on loans. If the loans aren't paid back, the bank not only loses the unpaid amount of the loan, but also the interest which would have been paid, and it's rare that the amount they'd get by repossessing a house and reselling it would exceed the unpaid principle plus the amount the back would have received in interest. A foreclosure usually means a loss for the bank.
Tuppence invested for 20-some years would be more likely to result in bus fare than pay off the mortgage on a house. As such, some viewers have speculated that Mr. Dawes Jr.'s story about that investment is an excuse for the bank president to arbitrarily forgive Michael Banks' debt in view of his nephew's treatment of him, without it appearing to be an act of charity.
Mr. Wilkins' character is inconsistent. He is portrayed for most of the movie as a dishonest and heartless man, telling lies about Michael Banks' standing in the bank and destroying legal evidence in pursuit of his fraudulent goal. But then he refuses his underlings' offer to leave work early, and magnanimously stays at his post to midnight to give Banks every last chance to win. After Wilkins' scheme is defeated, he is quite gracious and shows a remorseful conscience.