I'm going to preface this by saying that I'm coming at from a Biblical perspective, so mods, please let me know if this isn't an acceptable answer. The short answer is that as the Gospels were written, more blame was shifted away from the Romans and onto the Jews, to the point that Roman responsibility for the Crucifixion can be assigned to a few bad apples in the Roman Legion.

Boston College indicates that the Gospel of Mark was written first in the 70s, and Mark describes Pilate, the Roman governor overseeing the Crucifixion, as just trying to keep the peace. He presents no charges against Jesus, and merely hands Jesus over to the Jews to "satisfy the crowd". Next, Matthew and Luke were written around the 80s or 90s, and present Pilate in a pretty similar manner. The only real difference is that Matthew a) describes Pilate's wife as having a dream about Jesus before His arrest and telling her husband to leave Him alone, and b) recounts Pilate washing his hands of Jesus' fate, casting the blame onto the Jews (quite an interesting approach for a Jewish man writing to a Jewish audience). In Luke's Gospel, he states that Pilate told the crowd he would punish and release Jesus, but would not kill Him, and only turned Jesus over to the crowd to mollify them out of a riot.

And then we get to John, the last of the Gospels, and we get a...not necessarily different, but much deeper picture of Pilate. Pilate immediately finds Jesus innocent of any crime, and tries to hand Him back to the Jews, but the Jews say they cannot put anyone to death. Pilate then has a conversation with Jesus about Heaven and truth, and then attempts to have the Jews acquit him. When they do not, Pilate flogs Jesus and tells the Jews that he is releasing Jesus, since he does not see any crime worthy of a death sentence. When the Jews threaten to revolt, Pilate hands Jesus over out of fear, rather than indifference. He says that Jesus is the King of the Jews to the crowd, and seems to be defending Him out of a sense of reverence, rather than the innocence presented in the other Gospels.

The key difference with John is in a detail that all four Gospels share. As was Roman tradition, the Romans affixed a plank of wood to the cross of the condemned criminal with their crime written on it, and all four Gospels mention this plank and its inscription. John, however, is the only Gospel writer who states that the plank was written and affixed by Pilate, who was using it as a direct statement rather than the mockery presented in the other Gospels. The Synoptic Gospels say that "This is the King of the Jews" was used to mock Jesus, while John argues that it was a statement made by Pilate during the Crucifixion that Jesus was, in fact, King of the Jews. He even has a discussion with the chief priests about his statement, and Pilate gives his famous answer of "what I have written, I have written".

Can any of this be correlated to the Roman Empire's move towards Christianity? Most likely not. Christianity didn't become the state religion of the Roman Empire until 380 CE, under Theodosius, and even the most liberal historian would be hard-pressed to say that John's Gospel was written that late. According to PBS, Christianity didn't really begin to grow until the 2nd or 3rd Century, while John's Gospel was most likely written in the late 1st Century.

The shift of blame most likely happened to aid in the early church's evangelistic tactics. While the early church mostly consisted of Jews who saw Jesus as the fulfillment of Jewish prophecy, evangelism to the Jews was extremely difficult. Paul, who wrote the earliest parts of the New Testament, talked about the Mosaic Law and what aspects of Judaism new believers had to follow, if any. By the time we get to the Gospels, Christians had found that the Jews were unresponsive to the message of Jesus, but the Gentiles were loving it (see Galatians 3:28 and Colossians 3:11 for examples of the divide between Jew and Gentile being reconciled). This is where we can see a definite shift in the intended audiences of the Gospel writers. As mentioned, Mark is fairly even-handed, Matthew and Luke take a little away from Pilate and shift it to the Jews, and John presents Pilate as a man who recognized the truth but was bullied into handing over Jesus to prevent bloodshed.

I can't speak as much to the actions of Roman Emperors, since that's not my field of expertise, but the short answer is that yes, blame was primarily placed on the Jews, aided by the Gospels, with the actions of the Roman legionaries who mock and beat Jesus just being attributed to a few bad apples who didn't recognize the truth when they saw it.

Also another question that just popped into my head-- What did the Roman Emperors do about previous Emperors they had deified when they converted to Christianity-- Was it all forgotten overnight or did people still consider people like Julius Caesar as among the Gods?