Did emperor Nero later, any time before his death, regret anything that he did to the early Christians?
Is there any evidence that documents his opinions on his actions?
We don't know
Popular belief is that Nero was an incorrigable monster (Wiki, but more exemplary, the Christian Courier). However, history tends to be written by the victors, and Nero was ultimately the loser. Thus Edward Champlin in the New England Review suggests (emphasis mine):
But a very different and equally powerful image of Nero competed for favor down through the centuries. Rumors flew from he moment he disappeared, that he was not in fact dead, that he had fled to the East, gone into hiding and was now biding his time. He would return in triumph, like Arthur or Charlemagne, to slaughter his enemies, free his oppressed people, and reign again: some three and a half centuries later, in The City of Gody, St. Augustine writes precisely of the living, popular belief that Nero would one day be revealed and restored to his kingdom. A Jewish legend has him converting and becoming the ancestor of one of the great figures of rabbinic Judaism; even Christian stories have him listening respectfully to the teachings of Christ; at least three False Neros appeared and attracted large followings in the East; one, we are told, almost caused a war between Rome and the rival empire, Parthia. To fill in this astonishing portrait of a "good" Nero would take us too far off course into some curious byways of art, literature, and popular culture, but I do want to emphasize one aspect. Many years after his death, Dio of Prusa, no friend of Nero, claims flatly in one of his orations that the truth about Nero's death is unknown. Everyone, he adds even more startlingly, everyone wants him to come back, and most people believe that Nero is still alive. "Everyone wants him to come back." The truth is that outside of court circles and Christian congregations, Nero was vastly popular, both before and after his death. He was a popular monster.
The truth is that we may never know what Nero did in his later life. We don't even know precisely how he died. The horrors he perpetuated were undeniable, but that those descriptions have been biased in favor of the survivors is entirelly believable.
But I have a hard time believing, without remarkable historical evidence, that Nero regretted anything. To paraphrase a favorite line from the movie The Hunt for Red October, "there's little room in Nero's heart for anyone but Nero."
Nero ruled from 54 A.D. till he (allegedly) committed suicide on 9 June 68 A.D. Whether he took his own life because of remorse and regret will never be known. Judging from the historical records, he was a despot and a tyrant. Here is some information taken from a Wiki article:
Nero executed his mother and was probably responsible for the poisoning of his stepbrother Britannicus. Nero crucified Christians, threw them alive into pits with wild animals, and according to Tacitus, Suetonius and Cassius Dio, had captured Christians dipped in oil and set on fire in his garden at night as a source of living light. Nero was a despot and a tyrant and his persecution of Christians is a matter of historical infamy. The apostle Peter was crucified in Rome (some say upside down), the apostle John supposedly plunged into boiling oil (but suffered nothing) then banished to the island of Patmos where he was forced to work in the mines. Source: http://en.wikipedia.org/wiki/Nero
The article in the link below presents historical evidence to expose the cruelty of Emperor Nero with regard to the persecution and torture of Christians (after 66 A.D.). The details are too gruesome to copy and paste. However, the following brief extract illustrates the depravity of Nero:
For this spectacle Nero gave the use of his gardens, and appeared himself among the people in the garb of a charioteer, taking an active part in the Circusian games; himself standing in the circus, and, as charioteer, guiding a chariot. These proceedings, according to the testimony of Tacitus, although it had the appearance that the Christians were punished as malefactors who had deserved the extremest penalty, nevertheless moved the people to compassion; for they understood well enough that the Christians were not exterminated for the good of the common weal, but simply to gratify the cruelty of one man, Nero." Compare Abr. Mellin. 1st book van de Histor. der vervolg. en Mart printed Anna 1619 fol. 11. col. 4. and fol. 12. col. 1. with Tacit. Annal. lib. 15. and Tertul. Apol. Contr. Gent. cap. 50 and adv Marc. cap. 5. Martinal. Epig. 25. lib. 25.
Did Nero regret his cruelty to the Christians, as well as to his own relatives and any person who came between him and his naked ambition and lust for power? Does any psychopath feel guilt and remorse? I don’t know. More likely by far that when he realised the game was up he took pre-emptive action to prevent his enemies from tearing him to pieces (or worse).