The best way to answer this question is simply to go chronologically through the eschatological views of each era.

One of the earliest Christian sources we have is Paul. Most New Testament scholars these days agree that Paul believed that the end of the world was imminent, possibly even happening within his lifetime. In 1 Corinthians Paul advocates refraining from getting married since the world is ending so soon and it would be pointless:

Now about virgins: I have no command from the Lord, but I give a judgment as one who by the Lord’s mercy is trustworthy. 26 Because of the present crisis, I think that it is good for a man to remain as he is.27 Are you pledged to a woman? Do not seek to be released. Are you free from such a commitment? Do not look for a wife.28 But if you do marry, you have not sinned; and if a virgin marries, she has not sinned. But those who marry will face many troubles in this life, and I want to spare you this. 29 What I mean, brothers and sisters, is that the time is short. From now on those who have wives should live as if they do not; 30 those who mourn, as if they did not; those who are happy, as if they were not; those who buy something, as if it were not theirs to keep; 31 those who use the things of the world, as if not engrossed in them. For this world in its present form is passing away. —1 Corinthians 7:28-31, NIV

There are disagreements about what happened to Paul, but the most commonly accepted theory is that he was beheaded in Rome sometime in the 60s. Since he was not that far removed (temporally) from Christ’s life at the time of his death, it’s unlikely that his beliefs on the imminence of the apocalypse were ever challenged.

Shortly after Paul’s death, a catastrophic event occurred in Judea—the fall of the second temple. This event is difficult to interpret for a number of reasons. One is that some of our main Christian sources about it are the Gospels, which don’t speak of it directly, but purport to speak of it fulfilling a prophecy uttered 40 years earlier. The Gospels were most likely written after Paul’s letters and after the destruction of the temple. The destruction is presented within them as a prophecy (which the authors presumably already knew would happen, since they were writing after the fact, but the setting of the writing was much earlier). In several Gospel passages, Jesus presciently warns Judeans that no good will come from political revolution:

As Jesus was leaving the temple, one of his disciples said to him, “Look, Teacher! What massive stones! What magnificent buildings!” 2 “Do you see all these great buildings?” replied Jesus. “Not one stone here will be left on another; every one will be thrown down.” 7 When you hear of wars and rumors of wars, do not be alarmed. Such things must happen, but the end is still to come. 14 “When you see ‘the abomination that causes desolation’ standing where it does not belong—let the reader understand—then let those who are in Judea flee to the mountains. Mark 13:1-2;7;14, NIV

Jesus goes on in this passage to say that after this, God will gather God’s chosen ones and take them into Heaven, before claiming that “this generation will not pass away before these things happen.” ‘The abomination’ is a reference to a prophecy in Daniel 9:27, which predicts the placement of a slaughtered pig (an unclean animal in both modern and ancient Judaism) in the Holy of Holies, considered the most sacred part of the temple. This was done in 168 BCE, desecrating the temple (the full story can be read in the deuterocanonical books 1-4 Maccabees). The desecration of the temple predicted by Jesus in Mark involved Roman soldiers looting holy artifacts from the temple before its destruction. In Ancient Jewish theology, only a Jewish priest could enter the Holy of Holies so the presence of anyone else in that area would have been considered a desecration, especially a Gentile.

The destruction of the Temple and the tamping down of the rebellion in Judea in 70 CE was extremely violent. Josephus, our main source for these events, writes that “slaughter was everywhere.” (In his History of the Jews in Antiquity). The Gospel authors, who likely lived through these events, were hyper aware of this and it is possible that it is for this reason they chose to focus on (and guide their readers towards focusing on) the world beyond. The Romans would get their punishment soon, in the new age. There would be justice, in the new age. There would be freedom from Roman oppression, in the new age. Having seen the utter destruction wrought upon the Zealots (the main Jewish political faction responsible for the rebellion), many believed that justice on Earth was not possible and the only way justice could be achieved was in the next life when Christ returned.

For approximately the next 250 years, this was the prevailing mood amongst everyday Christians. Until Constantine, Christians were heavily persecuted throughout the Roman Empire. Each of these persecutions varied in terms of severity but they all led Christians to a strong belief that the end was near. The book of Revelation is widely considered to have been written in the 90s, during the reign of Domitian, as a coded response to the persecution Christians had endured under him, as well the severe suffering they had gone through under Nero (the infamous number 666 is a coded reference to Nero). Revelation again attempts to give Christians hope by claiming that judgment will come soon and there is still reason to hope for the future. Christ will return and “make all things new.” This hope in the face of intense persecution was the general attitude amongst Christians in the Roman Empire for a long time.

When Diocletian became emperor in the 290s, he reinstituted many of the persecution methods discontinued in the past. It is widely considered the most severe of all Roman persecutions of Christians. In the past, Christians had been forced to sacrifice to Roman gods or face imprisonment or fines. Diocletian ordered that Christian churches be burned, scriptures be confiscated, rescinded their legal rights, enslaved some of them, removed Christians from positions of power, and burned them alive.

During this time, among some Christians there was such a fervor to be a martyr that several Christian leaders wrote against the practice of becoming a martyr intentionally. This is evidence that many Christians still believed not only that Heaven was better than their current reality, but that their death could help usher in the apocalypse and ‘the new age’ promised in scriptures.

All this changed in 313 when Constantine did a full 180 and reversed the stance of the Roman Empire towards Christianity. Before, it had been illegal to be a Christian. Now it was legal. Later, it the official state religion (edit: I had originally written that Constantine made it illegal not to be a Christian. Several comments below pointed out that this was incorrect, so I’ve edited to reflect here).

In addition, Christians increasingly found political power in the new Roman Empire. This is the beginning of the Catholic Church’s eventual enormous influence over Europe, not only religious but political. Now the main concern wasn’t an evasion of suffering but how to best take advantage of the situation on Earth. The meshing of political power with religious identity made the issue of a belated eschatological reality a moot point. The most important thing now was the expediency being a Christian afforded someone within the empire and later in the various kingdoms of Europe.

If you’d like to know more, I’d recommend searching for articles or books on early Christian eschatology.

EDIT: Based on some of the comments below, I think I wasn’t clear enough on what I meant, so I’d just like to clarify. I’m not suggesting that joining of political and religious forces ended all persecution of Christians, nor that it made eschatological issues a non-issue. What I am saying is that prior to the legalization of Christianity, the imminent return of Christ had been a primary concern due to the situation many Christians found themselves in, whereas afterwards, it took a backseat to other issues like the nature of the Trinity and the Divinity of Christ. The legalization of Christianity and its subsequent role as state religion doesn’t mean Christians were now so tempted by political power that they just pretended not to care about the apocalypse anymore. It wasn’t my intention to imply this so I’m sorry if it came off that way. My argument is merely that the relaxed attitude of the Empire towards Rome allowed them not to be as concerned with this one issue at the expense of other theological issues, since it was no longer as pressing, as martyrdom and mass death were less likely.

Sources:

—Josephus, History of the Jews in Antiquity

—Paul: 1 Corinthiansand Romans

—Mark, Matthew

—Eusebius, Church History, Books VIII-X

—Michael Whitby, Christian Persecution, Martyrdom, and Orthodoxy (2006)

—Hans Schwarz, Eschatology (2000)

The other parent comment here does a good job of describing the overarching currents of history following the death of Jesus, but it contains some half-truths (in so far that they're inaccurate, overstate the evidence, or subscribe to a particular viewpoint) which I believe detract from its overall utility as a response to your prompt.

There is an extent to which an all-encompassing answer would be quite interdisciplinary, as your question makes a Christian theological assumption regarding the interpretation of canonical Scripture, namely the Olivet Discourse in Matthew 24. The idea given here follows the Apocalypiticist approach, popularized by Albert Schweitzer, that Jesus's statement that "This generation shall not pass away until these things come to pass..." was a prophecy emblematic of the Jewish Apocalyptic tradition. This idea is bolstered by and borne out of the modernist school of thought & the historical-critical method of exegesis. It comes as a methodologically natural answer to something that purports to be supernatural.

To say that most scholars now agree on these ideas is a bit of an obfuscation. A large number of Biblical historians, who attempt to reconstruct physical, causal explanations for occurrences have come to a logically coherent narrative and timeline that fits into this broader framework. To question why Christians continued to follow the words of Jesus even though the world did not end within a generation is a matter of conjecture that more neatly fits into anthropology or the history of philosophy.

One possible explanation is that these apocalyptic ideas were of only tangential concern to the lay Christian. In the grand scheme of things, the teachings of Jesus nicely complemented existent Hellenic and Judaic ideologies. As a faith, this amalgamation of Second Temple Judaism and Classical Greek thought was in its infancy- in that light, it became an ad hoc matter to establish a rigorous orthodoxy. This period of confusion can be highlighted by the Incident at Antioch (Galatians 2:11-14) in which Paul, a Hellenised Roman citizen who is pushing Christianity in a Gentile direction, butts heads with the representative from Jerusalem who wanted to maintain Jewish practices. This ideological split within proto-Orthodoxy persisted for centuries, and we haven't even mentioned the Gnostics or the followers of Mani.

While early Christians did often emphasize martyrdom [1], it was often in the context of a rejection of the imperfect physical world, not an attempt to bring about the end of days or to immanentize the eschaton. Further, there is an increasing body of scholarship that indicates that Roman persecution was scattered (carried out by local magistrates rather than at the advisement of the emperor) and largely a post hoc Christian fabrication [2]. By the time we get to the aptly-named Justin Martyr (c. 100-165), Christian theology is taking on traits of allegorical interpretation (a la the Alexandrian School) and proto-Augustinian illumination, with Christ operating as an explanation of how truth is ascertained (as a vision of God through the incarnate God who divinizes human flesh) [3].

So, why did Christianity persist? We're left with several choices. Ultimately, it's likely that it was simply the result if its own inertia. In this case, it's in many ways an accident of history. Secondly, it was philosophically useful for the state of metaphysical questions at the time. It was borne out of a necessity for explanation where there was none. Alternatively, we reject either the Apocalypiticist approach in particular or the naturalistic framework in general, at least perhaps in this case. But that last option is probably attractive only to a certain type of person.

[1] Chapter 6 of the Epistle of St Ignatius to the Romans.

[2] Shaw, Brent (2015-08-14). "The Myth of the Neronian Persecution". The Journal of Roman Studies. 105: 73–100. doi:10.1017/S0075435815000982

[3] Justin Martyr, Dialogue with Trypho