Barsad and Cly are not punished because Darnay and the courts have no interest in continuing the trial process.
The courts are quick to accuse traitors, but less interesting in pursuing conmen. Darnay has no interest in pursuing the matter, because he wants to keep a low profile. He is the nephew of the infamous Marquis St. Evremonde and the heir to the title, so there is definite trouble for him if he stirs up much attention.
Since “putting to death was a recipe much in vogue with all trades and professions,” it should not be a surprise that opening a letter would be considered a mortal sin worthy of death. Roger Cly and John Barsad are the witnesses against Charles Darnay.
John Barsad is a perfect gentleman. Who could question his word?
The story of his pure soul was exactly what Mr. Attorney-General had described it to be—perhaps, if it had a fault, a little too exactly. Having released his noble bosom of its burden, he would have modestly withdrawn himself … (Book 2, Chapter 3)
As honest as he may seem, Barsad is actually sneaky. His real name is Solomon Pross, and he is a conman and spy for hire. Barsad, the “hired spy and traitor” is accompanied by Roger Cly.
Roger Cly was Charles Darnay’s servant, and supposedly was loyal to him. Of course, he seems ready to testify against him.
The virtuous servant, Roger Cly, swore his way through the case at a great rate. He had taken service with the prisoner, in good faith and simplicity, four years ago. (Book 2, Chapter 3)
Cly’s no saint either. He faked his death and is able to make his way back and forth from France to England, making him a valuable spy.
Despite his apparent drunken stupor, Sydney Carton saves the day. All he has to point out is the fact that he and Charles Darnay look very much alike, and the jury is flummoxed. They acquit, and that is the end of it. Darnay has no desire to open up the can of worms and accuse the two men of bearing false witness against him. If he does, it could drudge up information about his real past that would cause trouble for him.
Likewise, the court system does not benefit for continuing. They made a mockery of the trial process, and they know it. Things are rough in France, and there are grumblings in the colonies. It is best to sweep the whole thing under the rug and forget about it before it becomes even more of an embarrassment.