One of the best ways to learn about a particular era of a country is through literature like A Christmas Carol, and the Carol can be looked at as an incredible historical piece, for it takes place during the Victorian era which occurred between 1832 and 1901.
We know that the novella was published in 1843 because Dickens tells us. We know that Dickens' style of writing falls under realism, the use of facts of his time combined with fiction, so we can assume that any references to life in Victorian England in the novella can be associated with that date. So let's begin our search for historical references to 1843 Victorian England.
The first bit of evidence would be Scrooge's name, for it was a colloquial or vulgar word of the time which meant to crowd or squeeze. There were a few variants of the word's spelling during the time such as scroodge, scrowde, scrowge, and skrouge.
Speaking of words, another word that Dickens uses that gives authenticity to the 1843 date is the use of the word "nuts' in the following passage from "Stave I":
But what did Scrooge care? It was the very thing he liked. To edge his way along the crowded paths of life, warning all human sympathy to keep its distance, was what the knowing ones call "nuts" to Scrooge.
The use of "nuts" in this passage meant "good luck" in 1843.
The third bit of evidence that A Christmas Carol takes place in 1843 is found in the reference to the Union workhouses. These workhouses were established by the Poor Law of 1834 which stated that two or more parishes unite to provide a home for the destitute where they might labor in exchange for their room and board.
The diction and the references to governmental agencies in A Christmas Carol do support its 1843 setting.
I don't know that there's anything that can actually prove that the setting is in 1843 instead of, say, 1844. But there are plenty of things that can prove that the story is set in England of around that time.
We can tell it's in England because they are using English money ("half-crowns") and talking about places in England ("ran home to Camden Town").
We can tell the general time period. The people (who can afford it) are going around in carriages. Others are offering to move ahead of the carriages carrying torches to light them on their way. But it's not too long ago because men are working on gas pipes.
In the original version of "A Christmas Carol" there are several clues that indicate that the setting takes place in England. In Stave 1: The story reads
"Saint Paul's Churchyard" which is found in England.
Moving through the Stave pages we also see on page on a discussion by Scrooge with his nephew in which he talks about Parliament, the English government. The nephew indicates to Scrooge that because Scrooge is such a good speaker, he is surprised that Scrooge as not gone to Parliament to address his issues.
Scrooge questions two men who are collecting for the poor by asking;
"The Treadmill and the Poor Law are in full vigour, then?" said Scrooge.
These are laws that were enacted in England and resulted in the incarceration of people unable to pay their bills.
The tale makes mention of The Lord Mayor, which was a position occupied in towns in England. Another line that directly makes mention of the location is;
"about him as any man in the city of London"
There's a reference to a "United States security." There was an economic panic in 1837 (I think) in the US and hundreds of banks failed. Obviously, just like today, this would have cuased people like Scrooge to lose money because they bought those securities and now they were worthless.