How was the Church able to handle it, and even grow exponentially, when this didn't happen?

1 Thessalonians addresses exactly this problem: What do you say to Christians who are disappointed that Jesus hasn't yet returned? In chapter four, Paul addresses the Thessalonians' concerns about “those who sleep in death” (4.13). It appears (based on Paul's reply; we don't have the letter to which he is responding) that the Thessalonians had been given the impression that Christ's second coming was imminent; when members of the community died before Christ's return, the survivors worried that their fallen brothers and sisters would therefore be excluded from the new kingdom. Paul reassures them that

we who are still alive, who are left until the coming of the Lord, will certainly not precede those who have fallen asleep. For the Lord himself will come down from heaven, with a loud command, with the voice of the archangel and with the trumpet call of God, and the dead in Christ will rise first. After that, we who are still alive and are left will be caught up together with them in the clouds to meet the Lord in the air. And so we will be with the Lord forever (4.15-17).

In other words, there are contingencies in place for those who succumb to death before the end of the world. There is a plan, Paul reassures them, for an extended wait.

Paul also makes the timeline a little more vague. In the next chapter, he reminds the Thessalonians of a particular aspect of his earlier teaching, telling them that “about times and dates we do not need to write to you, for you know very well that the day of the Lord will come like a thief in the night” (5.1-2). That is, no one knows when the end will come, as like a thief it will come by surprise. This might strike you as somewhat surprising itself, as speculation about the end of the world typically includes all kinds of signs that indicate when the end is nigh (cf. 2 Thessalonians 2, where Paul offers a number of signs (one reason some scholars doubt its authenticity)). Why would Paul preach to the Thessalonians that the exact timing of the end of the world is impossible to predict? Well, if the world does not end when he says it will, he's going to have a lot of disappointed people. Better to be vague. This is sometimes referred to as “delayed eschatology:” The end of the world is DEFINITELY coming, but it's coming at some ill-defined point in the future. That way, Christianity can have the end of the world as central to its beliefs without it dominating present considerations.

There were many other strategies employed, but these (& the realized eschatology of Luke) are the earliest & some of the most influential.

At what point was this no longer the predominant expectation of new converts?

It's hard to say what the majority of early converts believed, as we simply don't have access to that sort of data. What evidence we do have –- a handful of 1st century Christian texts –- doesn't express too much interest in imminent eschatology. As 1 Thessalonians suggests, even the earliest converts were dissuaded from thinking of the end of the world as something that would occur soon. Similarly, in 1 Timothy there's lots of talk about establishing a church hierarchy, rearing children, planning for the future, etc. You wouldn't expect Christians to be doing those sorts of things if they thought the world was going to end next week.

That said, there's never been a point in Christian history when someone wasn't preaching that the world was about to end. Indeed, this sort of message is especially effective at attracting converts. The problem is that imminent eschatology doesn’t lend itself to the creation of the sort of enduring institutions that a religion needs to grow and prosper. There's an excellent modern example of this in Seventh-Day Adventism (see Lawson, “The Persistence of Apocalypticism Within a Denominationalizing Sect”): Converts are drawn in with a promise of imminent destruction, but comparatively little is said about the end of the world among established members of the church. It's difficult to get people to plan and build and foster community if they think God is about to wipe everything away.

Rather than think of imminent eschatology as a Christian belief that was superseded, then, it might be better to think of it as a persistent Christian belief that is specifically targeted at potential converts. Telling people that the world is about to end is a great way to scare the crap out of them and get them to join, but the messaging usually shifts once they're on the inside. Hook 'em with the fire and brimstone, then give them their assignment for next week's bake sale.