Paul's Untimely Birth?
I will use the concordance-friendly KJV to answer this question. Notice that in 1 Cor 15:1-4 Paul is speaking expressly declaring the gospel--Jesus' death, burial, and resurrection in full accordance with the scriptures.
Beginning with verse 5, Paul is giving eye witnesses to the resurrected Christ--telling about those who saw the resurrected Christ with their own eyes. Twice he mentions one of the apostles singly (verses 5 and 7). Twice he mentions the all the apostles because we know that Christ appeared to them more than one time as a group (verses 5 and 7 also).
This question appropriately arises out of the very next sentence because of the immediacy of need to explain his connection to the apostleship group, and the necessity of proving that he was also an appropriate eye witness of Jesus' resurrection. This requirement was revealed to be the necessary element by Peter in seeking a replacement (at least in number) for Judas, in Acts 1:21-22:
Wherefore of these men which have companied with us all the time that the Lord Jesus went in and out among us, Beginning from the baptism of John, unto that same day that he was taken up from us, must one be ordained to be a witness with us of his resurrection. (My emphasis)
This appears to leave out Paul as far as being eligible to be an apostle-OR DOES IT? Some say that because Paul had not "companied with" the other apostles "all the time", beginning with the baptism of John, unto the day that Jesus was taken up from the other apostles, he could not have been an "apostle of Jesus Christ". Yes, Peter and the others "appointed" two men as candidates, casting lots on them to see which one was the lucky, or unlucky one Acts 1:23-25). The lot fell upon Matthias in Acts 1:26:
And they gave forth their lots; and the lot fell upon Matthias; and he was numbered with the eleven apostles.
Therefore, the apostleship of Paul needed an explanation at that very point in Paul's description of the eye witnesses of Christ's resurrection.
verse 8 in the KJV states:
And last of all he was seen of me also, as of one born out of due time.
Here, Paul is letting us know that he was a son of God through faith--just like the other apostles-- and even more, that he was indeed an apostle of Jesus Christ. Notice the explanation of OP's verse in question in the very next three verses, verse 9-10:
For I am the least of the apostles, that am not meet to be called an apostle, because I persecuted the church of God. But by the grace of God I am what I am: and his grace which was bestowed upon me was not in vain; but I laboured more abundantly than they all: yet not I, but the grace of God which was with me. (My emphasis)
Being "born out of due time" appears to be a very precise term--more so than merely, "untimely birth". Untimely birth could mean a premature birth, as well as a late birth. However, "due time" had already passed (according to Peter and the other apostles) for the birth of Paul when he was stopped on the road as he was nearing Damascus (See Acts, chapter 9). At that time, Paul was still warring with those disciples of Christ, including the very apostles he had just mentioned in verse 5 and 7.
The importance of this is that, notwithstanding whatever contentions made--pro or con--about how, or if, Paul met Peter's criteria for being an apostle of Christ, it is Christ who chooses the ones who he sends, not Peter, not anyone else. Acts, chapter nine clearly reveals that Christ chose Paul to do EXACTLY what the other apostles did, as beautifully expressed in verse 11:
Therefore whether it were I or they, so we preach, and so ye believed. (My emphasis)
Maybe Paul was late compared to the others in his being "born again", but his message was identical to that of the other apostles--"so we preach, and so ye believed." Clearly, he was "sent by" Christ Himself, with orders, not Peter, not the other apostles (see: apostolos-ἀπόστολος-Strongs G6652). Paul was truly an apostle of Christ, sent to declare the gospel, just like he began to declare in 1 cor 15:1:
Moreover, brethren, I declare unto you the gospel which I preached unto you, which also ye have received, and wherein ye stand;