I'm interested in how they responded to this change, especially when it coincided with such an interesting time in Jewish history (the destruction of the Temple and the solidification of the diaspora). Were there any significant developments, theological, legal, or otherwise when the communities Jews found themselves in transformed into Christians, who may not be warm to Judaism, but may be distinguished from general pagans?
The change was not as drastic as you might expect. The Roman Empire became Christian pretty gradually. Plus, a pretty large share of Jews at the time lived in the Middle East. The big thing is, the Jewish communities that were writing the most volumously were middle eastern. Jews in ex-Roman areas didn't really write a whole lot until the Middle Ages. Particularly, the center of Rabbinic thought was concentrated in the Middle East during that period. Our biggest source of Rabbinic literature is definitely from Babylonia, far removed from Europe.
We do know that Christian Rome did abolish some of the Jewish community's formal autonomy was abolished, i.e. disputes between Jews only went before a Jewish court if the litigants agreed. Previously, Jewish law had been an alternate legal system under Roman authority. Basically, between Jews, Jewish law, adjudacated by Jewish courts, was the legal system. With Christianity becoming the dominant Roman religion this ended. But still, Jewish communities had a high degree of autonomy, since the authorities tolerated it and the Jewish community itself maintained it. This general arrangement was maintained for the next 1500 years or so.
However, the theology of Judaism did not change drastically with the rise of Christianity. Christianity did influence Judaism in some ways, but the rise of Christianity as the Roman religion didn't have a huge impact on Judaism.