I have read the books multiple times but not seen the entirety of the TV Show even once. The reasons Jaime never tells anyone about the wildfire plans is-
1. It is pointless without definite proof
Everyone who likes/supports him will believe this and think him a hero; everyone who dislikes him or wants to see the Lannisters fall will take it to be a self-serving lie. His father had defected from the Targaryen side AFTER the war was almost decided. So his actions would have looked opportunistic. This refutes:
"But had he spoke, Jaime would have been seen as a hero, not a villain. He would have been forgiven instantly for his decision to slay the king. For a society that values honor, they would understand Jaime was torn between an oath he made and the desire to save half a million lives."
Varys may not have corroborated his story. It is quite possible nobody heard Aerys' words as that was said to Rossart not in normal court.
The same happens regarding the 'truth' content of Stannis' proclamation throughout the realm-
Catelyn had a glimmer of understanding. "Stannis means to parade his brother's bastard before the realm, so men might see Robert in his face and wonder why there is no such likeness in Joffrey."
"Would that mean so much?"
"Those who favor Stannis will call it proof. Those who support Joffrey will say it means nothing."
A Clash of Kings, Catelyn VI
Also, while Targaryens are known to be a bit mad, they still hold love/respect in many houses and by many smallfolk through Westeros (known example-Darry).
2. The Oath
Honorable lords like Ned Stark (and most other lords if they wanted to be politically correct) would have expected Jaime to at least not kill the King he was sworn to defend. The urgency of Jaime's situation (to protect the city from burning in wildfire) would not have mattered to them in hindsight.
3. Jaime's youth and nature
Jaime never claimed to be righteous, he did what he thought was right and what was his natural instinct in killing the King but he never claimed to care about anyone but his family. Why would anyone believe he wanted to protect the city from wildfire. Jaime is always portrayed as impulsive in everything, so his actions during this time also were impulsive and not the result of reasoned argument.
4. Issue of loyalty
Even knights like Arthur Dayne stayed loyal to the Throne despite Aerys II's madness. So it would be hard to believe his actions were not merely opportunistic and were (partially) forced. (He did defect by choice but he had good and sufficient reasons according to his thinking).
Also it may be that he did not know who will win the fight at Kings Landing so if Aerys II had been just imprisoned/held alive till then someone may have saved him/helped him escape (Varys?) and the whole realm would have bled in answer (including the Lannisters, who Jaime loved). This happened after Tywin revealed his allegiance in the Sack of King's Landing.
Any reason Jaime would have given would have sounded like an excuse, and would have implied that Jaime's actions were wrong (which he did not believe they were), but why would a proud Lannister give any excuse to anyone else, let alone a Stark.
That leaves only personal contacts for Jaime to tell the truth to-
Kingsguard- Only Kingsguard alive (Ser Barristan) never asked for Jaime's side of the story and judged him to be an oathbreaker (which he was,to be fair) so there was no reason for Jaime to tell him the truth. Or any of the new Kingsguard (Jaime also considers them inferior). I expect he may have spoken to Arthur Dayne (who seems to be the only father-like figure who he really respected).
Cersei/Tyrion- Maybe he did tell them, we really can't say from the books.
Tywin- It is possible Tywin never asked Jaime if he had a reason. He may have expected Jaime to act in a way that was best for the Lannisters and killing the king worked out fine with Cersei becoming queen.
The book does not provide with any insights other than the quote mentioned in TheLethalCarrot's answer.
He only had to look at me to judge me guilty." Jaime lurched to his feet, the water running cold down his chest. "By what right does the wolf judge the lion? By what right?"
A Storm of Swords, Jaime V