The chapters describe two different “fellows” and their feelings for Lucie Manette and manner of addressing her. The titles are ironic because the “fellow” discussed in the chapter is actually the exact opposite of the characterization the title describes. Specifically, "The Fellow of Delicacy" talks about Mr. Stryver, an indelicate, pompous fool who has no understanding of how he is perceived by others. "The Fellow of No Delicacy" describes Sydney Carton, who (somewhat ironically given his lifestyle) is a sensitive and feeling person who has tremendous delicacy when it comes to confessing his feelings to Lucie.

For example, chapter 12 ("The Fellow of Delicacy") opens with:

Mr. Stryver having made up his mind to that magnanimous bestowal of good fortune on the Doctor's daughter, resolved to make her happiness known to her before he left town for the Long Vacation. After some mental debating of the point, he came to the conclusion that ... he should give her his hand a week or two before Michaelmas Term ...

Several elements of this opening paragraph belie the description of Mr. Stryver as a fellow delicacy. First, he has made up his mind without any input from Lucie about whether she would be receptive to a marriage proposal from him. Dickens even says, "As to the strength of his case, he had not a doubt about it."

Second, the proposal is described as "that magnanimous bestowal of good fortune" because Mr. Stryver believes that Lucie would be lucky to have him for a husband. Third, he resolves to let her know about her pending happiness (more irony) because he is making plans under the assumption that Lucie will acquiesce to all that he proposes, from marrying him to when they will marry.

By comparison, Sydney Carton truly has feelings for Lucie. Moreover, he understands himself in a way that elicits pity from Lucie. When Lucie delicately suggests that Sydney change his lifestyle, Dickens writes:

Looking gently at him again, she was surprised and saddened to see that there were tears in his eyes. There were tears in his voice too, as he answered:

"It is too late for that. I shall never be better than I am. I shall sink lower, and be worse."

Sydney understands his limitations and, in fact, overestimates them. Sydney Carton could be someone who would be of interest to Lucie if he had the ability to give up drinking and change other aspects of his profligate lifestyle. Mr. Stryver, on the other hand, has little to no self-awareness. He believes that Lucie would welcome a marriage proposal from him and overlooks his shortcomings.

The titles are ironic because they express their meanings in terms which suggest the opposite. Far from being delicate, the pushy, headstrong, boorish Mr. Stryver is anything but. A self-made man who worships at the altar of his creator, Stryver is firmly of the belief that he is God's gift to women and that any woman in her right mind would be positively delirious at the prospect of marrying him, including Lucie Manette.

These character defects are bad enough, but Stryver also lacks any sense of tact or diplomacy, which stands in stark contrast to Mr. Lorry, who is the real "fellow of delicacy" in this particular chapter. For it is Lorry who manages to convince Stryver that he should go and see the Manettes and find out what they think about Stryver's plans to marry Lucie.

In the following chapter Sydney Carton shows the kind of delicacy of which Stryver can only dream. In expressing his love for Lucie, he does so in a gentle, unassuming way, almost apologetically. But the content of what he has to say is far from delicate, expressing as it does Carton's profound self-loathing as well as touching on the dissolute nature of his lifestyle. In that sense, one could say that Carton is indeed "a fellow of no delicacy," though not quite in the way that we'd expect.

Absolutely ironical, the two chapters in Book the Second provide character descriptions of C. J. Stryver and Sydney Carton.  While the reader would expect Stryver, who "shoulders his way" through life, knocking people about at Tellson's Bank and who exploits the talents of his associate, to be "The Fellow of No Delicacy," his character development takes place in the chapter entitled "The Fellow of Delicacy."  Furthermore, Stryver adds to his quality of obtuseness by informing Mr. Lorry that he has decided to offer himself as a husband to Miss Manette as Dickens wrily narrates,

Mr. Stryver having made up his mind to that magnanimous bestowal of good fortune on the Doctor’s daughter, resolved to make her happiness known to her before he left town for the Long Vacation.

To his announcement that he is going to marry Lucie--"no plainer case could be"--even before he has spoken to her, the astonished Mr. Lorry can only remark, "Oh, my!"  And, when Mr. Lorry tries diplomatically to dissuade Mr. Stryver, the crass Stryver demands of Lorry what is the meaning of his words.  After becoming angry at Mr. Lorry's words that he should not go to the Manette's without having some cause to believe that he should succeed.  Enraged at the insult, Stryver, ironically called "the fellow of delicacy" retorts,

"...You deliberately advise me not to go up to Soho and offer myself--myself, Stryver of the King's Bench bard?....ha, ha!--beats everthing past, present, and to come."

 Stryver projects the inadequacies upon Lucie, declaring her "a mincing fool."Then, the "delicate" Mr. Stryver turns and bursts out of the Bank, blowing past two clerks who hung onto their counters for dear life.

Unlike the bullish Stryver, Sydney Carton is quiet and passive.  Lacking confidence in himself, he,nevertheless, "carries his delicacy into Devonshire to the home of the Manettes, but he cannot bring himself to propose to Lucie as he feels unworthy of her.  However, he does pledge his loyalty and love to her, but in a subtle manner.  As a "fellow of no delicacy," Carton ironically is the man who speaks more deferentially to Lucie than any other.  Humbly, this truly delicate man, not the fellow of no delicacy as he is called, tells Lucie,  

"I have had unformed ideas of striving afresh, beginning anew, shaking off sloth and sensuality, and fighting out the abandoned fight. A dream, all a dream, that ends in nothing, and leaves the sleeper where he lay down, but I wish you to know that you inspired it.”

 Carton avows to Lucie that he will do all in his power to protect her and those that she loves.  Rather than announcing to Lucie how privileged she would be to have him as a husband as Stryver has planned to do, Carton humbles himself, saying he is not worthy to do anything but serve Lucie with his love:

"I have had unformed ideas of striving afresh, beginning anew, shaking off sloth and sensuality, and fighting out the abandoned fight. A dream, all a dream, that ends in nothing, and leaves the sleeper where he lay down, but I wish you to know that you inspired it.”

Clearly, the titles of Chapter 12 and 13 of Book the Second are ironically switched.