It is almost universally accepted that the first epistle of Saint Clement was written in the last year of Domitian or in the first year of his successor (i.e. A.D. 95-96).
It's interesting to see which of Paul's epistles Clement cites.
Clement 2:7 cites Titus 3:1 (one of the alleged pseudepigraphical epistles of Paul). This would mean that by the 90s, Christians in Rome were already circulating the Pastoral Epistles, so that they were read and memorized.
Clement 5:2 alludes to Galatians 2:9.
Clement 34:8 alludes to 1 Cor 2:9.
Clement 36 is full of quotes from Hebrews.
Clement 44 seems to be aware of Paul's instructions in 1 Tim and Titus.
Clement 47:1-2 cites Phil 4:15 directly as "the epistle of the blessed Paul the Apostle."
Clement 61:2 alludes to 1 Tim 1:17.
From these citations, we see that Clement has Galatians, 1 Corinthians, Philippians, Hebrews, and he also likely had 2 Timothy and Titus.
This reveals that already by A.D. 96, the epistles of Paul (including Hebrew, 1 Tim, 2 Tim, and Titus) were being circulated. The Paul corpus had been brought together. 2 Peter testifies to the same.
The question remains, who brought together the corpus of Paul's work?
I believe that it was Paul himself. For example, we have the letters of Cicero not because someone knocked on the door of every person to whom Cicero sent a letter. Rather, Cicero had a copy made of every letter that he wrote and collected them. Thus, at his death, Cicero had created the collected works of Cicero simply by virtue of his keeping record of his correspondence.
We know that Paul did something similar: "When you come, bring the cloak that I left with Carpus at Troas, also the books, and above all the parchments." (2 Tim 4:13).
Thus, when Paul died, the Pauline corpus was already assembled and ready for copying, if it were not already being copied in his own life time.