Who assigned the gospel authors originally? According to the theory of the originally anonymous gospels, the titles of the most ancient surviving manuscripts were added later on.

I would like to know:

For each gospel, how many different, distinct, "assigners" were there? When did they do the assignment and where were they? I am not asking about the authorship of the gospels, but supposing there were no titles, who assigned them.

For example:

The gospel according to Matthew was assigned by:

1) X1 person or group, in the Y century in the Z region.

2) X2 person or group, ...""

If much of this is not possible at least list the number of the different assigners and whether they may have assigned these texts independently of one another.

The gospel according to Luke... ""

Please give sources, thank you!
One of the earliest "assignees" so-called of the same names of Matthew, Mark, Luke, and John to their respective gospel accounts is Irenaeus of Lyons (~130-202 AD) who was a well-known, respected, apologetic writer about Classic Christianity.

Here is his account of the 4 gospels and his reasonings.


  
  It is not possible that the Gospels can be either more or fewer in number than they are. For, since there are four zones of the world in which we live, and four principal winds, while the Church is scattered throughout all the world, and the "pillar and ground" of the Church is the Gospel and the spirit of life; it is fitting that she should have four pillars, breathing out immortality on every side, and vivifying men afresh. From which fact, it is evident that the Word, the Artificer of all, He that sitteth upon the cherubim, and contains all things, He who was manifested to men, has given us the Gospel under four aspects, but bound together by one Spirit. As also David says, when entreating His manifestation, "Thou that sittest between the cherubim, shine forth." For the cherubim, too, were four-faced, and their faces were images of the dispensation of the Son of God. For, [as the Scripture] says, "The first living creature was like a lion," symbolizing His effectual working, His leadership, and royal power; the second [living creature] was like a calf, signifying [His] sacrificial and sacerdotal order; but "the third had, as it were, the face as of a man,"-an evident description of His advent as a human being; "the fourth was like a flying eagle," pointing out the gift of the Spirit hovering with His wings over the Church. And therefore the Gospels are in accord with these things, among which Christ Jesus is seated. For that according to John relates His original, effectual, and glorious generation from the Father, thus declaring, "In the beginning was the Word, and the Word was with God, and the Word was God." Also, "all things were made by Him, and without Him was nothing made." For this reason, too, is that Gospel full of all confidence, for such is His person. But that according to Luke, taking up [His] priestly character, commenced with Zacharias the priest offering sacrifice to God. For now was made ready the fatted calf, about to be immolated for the finding again of the younger son. Matthew, again, relates His generation as a man, saying, "The book of the generation of Jesus Christ, the son of David, the son of Abraham; " and also, "The birth of Jesus Christ was on this wise." This, then, is the Gospel of His humanity; for which reason it is, too, that [the character of] a humble and meek man is kept up through the whole Gospel. Mark, on the other hand, commences with [a reference to] the prophetical spirit coming down from on high to men, saying, "The beginning of the Gospel of Jesus Christ, as it is written in Esaias the prophet,"-pointing to the winged aspect of the Gospel; and on this account he made a compendious and cursory narrative, for such is the prophetical character. And the Word of God Himself used to converse with the ante-Mosaic patriarchs, in accordance with His divinity and glory; but for those under the law he instituted a sacerdotal and liturgical service.
  -AH III XI 8-
  


In addition from this, the same gospel names have survived to this day.

I am not aware of any other names assigned to the four gospel accounts besides these four until perhaps circa 1800 as questions from the Tubingen School of Theology arose specifically about why John's gospel supposedly contradicts the Synoptic Gospels (Matthew, Mark, Luke) account of the days and dates of the death, burial, resurrection of Christ Jesus.  Because of that so-called contradiction, ideas that John's gospel really wasn't written by John arose.