Acts 14:21-23
[21] And when they had preached the gospel to that city and made many disciples, they returned to Lystra, Iconium, and Antioch, [22] strengthening the souls of the disciples, exhorting them to continue in the faith, and saying, “We must through many tribulations enter the kingdom of God.” [23] So when they had appointed elders in every church, and prayed with fasting, they commended them to the Lord in whom they had believed.
After preaching the gospel in Derbe, Paul and Barnabas returned to the churches of Lystra, Iconium and Antioch where they, among other things, appointed elders and commended them to the Lord after praying and fasting, indicating that these churches were established and operational prior to having any elders appointed.
Titus 1:5
[5] For this reason I left you in Crete, that you should set in order the things that are lacking, and appoint elders in every city as I commanded you—
Apparently Paul established a church in Crete where he left his disciple Titus to appoint elders in every city where there was a church. Though Paul was an apostle, he designated this task to his disciple.
I Timothy 1:3-4
[3] As I urged you when I went into Macedonia—remain in Ephesus that you may charge some that they teach no other doctrine, [4] nor give heed to fables and endless genealogies, which cause disputes rather than godly edification which is in faith.
Paul left Timothy in Ephesus in order to give direction to the church and apparently appoint elders according to chapter 3. The appointment of Elders was left to Timothy's discretion, probably at the recommendation of those within the church who knew the men better.
II Timothy 4:20
[20] Erastus stayed in Corinth, but Trophimus I have left in Miletus sick.
It's possible that Paul left Erastus in Corinth for the same reasons that he left Timothy and Titus in their respected cities, one of those being to appoint elders of the church he had established there.
I Thessalonians 3:1-2
[1] Therefore, when we could no longer endure it, we thought it good to be left in Athens alone, [2] and sent Timothy, our brother and minister of God, and our fellow laborer in the gospel of Christ, to establish you and encourage you concerning your faith,
Timothy was sent to Thessalonica to "establish them," which probably included the appointment of elders.
It seems that Paul, as an apostle, planted the churches, but left the appointment of elders to his disciples at a later time.
The apostle John refers to himself as "The Elder" in two of his letters to the churches which he likely established, however, to my knowledge, we are not told if he appointed additional elders to those churches or not.
The apostle Peter addresses the elders of the churches of Pontus, Galatia Cappadocia, Asia and Bithynia in his letter to these churches, and refers to himself as a fellow elder. He instructs the elders to shepherd the flock among them, willingly serving as overseers, not for dishonest gain but eagerly serving; not being lords over those entrusted to them, but being examples to them. The elders were not to be elevated in position above the rest of the church, but were to demonstrate servanthood to the church.
Even though elders may not have been immediately appointed to the churches established by the apostles, the apostles themselves were overseers of the churches at least until they were established and elders were appointed. This is evident by the letters to the churches which are provided in Scripture. At least some of these letters were written to correct the church of error and to warn them of antichrists and deceivers who were infiltrating the church and leading others astray with false doctrines. These matters were apparently brought to the attention of the apostles by members of the churches, as we see in 1st Corinthians 1:11 where we are told that members of Chloe's household informed Paul that there were contentions among them. Though there was not an appointed elder within the church at the time, there were members of the church who were looking out for the health of the church and reporting such matters to the apostles for direction and to address the issue. We see in 1 Corinthians 5 that Paul addresses one of the issues, but then leaves it to the church to actually implement the discipline. Even with the absence of elders within every church, there was still accountability for the churches and protection from antichrists and deceivers.