That’s gonna be a difficult question there, because first of all you can’t generalize things. Between those four centuries Christianity was gradually spreading into different places, and the situation was different depending on the area.
Let’s start with Judaea. Eusebius reports a tradition of Christians in Jerusalem migrating into
Pella during the Jewish-Roman War of AD 66-70 (
HE 3.5.3). As always there are those who question just how much of Eusebius’ story here is based on fact, but if it did happen, it might explain why the position of the Church in Jerusalem dwindled, leaving only the Churches in Asia Minor and Europe: without a visible entity to speak with authority, its respected position as a guide would begin to fade.
Note that before AD 70 there wasn’t really any major distinction between ‘Christianity’ and ‘Judaism’. First-generation Christianity (aka the ‘Jesus Movement’ or ‘The Way’) was simply one of the different Jewish sects (some would even say ‘Judaisms’) that flourished during the Second Temple Period; Christians were simply a group within Judaism like the Pharisees, the Sadducees or the Essenes. Christian Jews, as faithful religious Jews, regarded “Christianity” as an affirmation of most every aspect of contemporary Judaism, with the addition of the belief in Jesus as the Messiah.
Of course, by this time, Christian Jews still worshipped in the Temple (cf. Acts 2:46; 3:1; 5:42; also the saying in Matthew 5:23), but at the same time they viewed it as a doomed institution (Jesus’ prediction of the Temple’s destruction is one of the well-attested strands of gospel tradition; Matthew 23:38; 24:2; 26:61; 27:40; Mark 13:2; 14:58; 15:29; Luke 13:35; 19:44; 21:6; John 2:19; cf. also Acts 6:14; Gospel of Thomas 71); they participated in it while it lasted, but they apparently did not expect it to last long. Not to mention that the Jerusalem Church saw itself as the new, eschatological Temple.
From the view of the community itself as the new Temple there seemed to have been two opinions standing side-by-side: one is that Christians could (or should) worship in the Temple while it still stood (that is, until God removed it), and the other, that since the Temple was already superseded by the Church, Christians need no longer participate in it. The Church in Jerusalem seem to have held the former view - because of this it maintained some standing in the Jewish community, even if with some difficulty (cf. the advice given to Paul in Acts 21:20-25). The author of Hebrews, by contrast, takes the latter opinion. St. Paul seemed to have been somewhere in the middle: the Temple may have no longer mattered to him as well, but he had no qualms about participation in Temple worship (cf. Acts 21:26) under the principle of becoming a Jew in order to win Jews (1 Corinthians 9:20). Since the Temple was an integral part of ‘Jewishness’, where Christians were perceived to be treating the Temple the way the Samaritans did, nothing was more calculated than to treat them as schismatics and a threat to Jewish identity. Hence controversy about the Temple was the occasion for the first major persecution of Christians (Acts 6-8). A way to dispel suspicions was of course for Christians to demonstrate their loyalty and reverence for the Temple in Jerusalem.
The Temple’s destruction in AD 70 thus had a dramatic impact on Christianity as a whole; it was when its distinctive character became more apparent. Now without a Temple the Church would be more free to proclaim its independence. Even since earlier, Christianity was already in the process of shedding some distinctively Jewish customs through moves such as rejecting the idea of circumcision as a requirement or modification of the Sabbath observance; with the Temple gone, they stood out all the more because they were not too concerned about it.
The issue of the Temple did not disappear after its physical destruction, because it did not cease to be central to Jewish identity. Few would have expected its loss to be permanent - after all, it was already destroyed once. Consequently, in Christian literature produced during the period between AD 70 and the Bar Kokhba revolt of the 130s, the issue is alive and well precisely in texts in which the growing schism between Christianity and common Judaism is clear and painful: the gospel of John and the epistle of Barnabas.