I attempted to address this question a while back (here). In short, the Romans didn't reconcile it, they didn't write about it or think about it in these terms. There is a decided lack of any writing where a Roman Christian struggles or contemplates a collective sense of Roman guilt as a corporate state-entity.

I would be very happy to come across an ancient text that did discuss this question though.

For the first part of the question: https://www.reddit.com/r/AskHistorians/comments/nlhj0/why_was_western_civilization_so_quick_to_accept/

In short: It started really slow, around Constantine and Theodosius (late 3rd to late 4th century) it became the main cult (but still not a majority, just the biggest of a lot of fractured cults) and then it still took a good two centuries, until it became the majority. What is not mentioned above is, how long it took until it was virtually prohibited to not be a Christian but I would guess another few centuries, probably around the time when the different Christian early sects consolidated into "Orthodoxy".

As for the second part of the question, I think this is rooted in a wrong assumption of national identity. "They" did not have to "reconcile their involvement" because they did not identify themselves as a bad guys in the Bible, simply because they both were people "of the Roman Empire". That's like saying that every European or (Colonist-) American person needs to reconcile with the atrocities during the Crusades.