The idea that human beings in the future kingdom of God will become spiritual beings similar or equal to angels is actually already present in the Synoptic Gospels, which were originally written in the late first century CE and are included as part of the Christian New Testament canon.

The Gospel of Mark (henceforth gMark), the earliest surviving gospel, which was most likely written sometime around 70 CE or thereabouts, portrays the Sadducees (i.e., followers of a particular sect of Second Temple Judaism who held that there is no life after death and that there will be no future general resurrection of deceased) as coming to Jesus to ask him the trick question: If a woman has had seven husbands in succession over the course of her life, all of whom have died, enabling her to remarry, to whom will she be married when God resurrects them all? Jesus replies in gMark 12:25:

"ὅταν γὰρ ἐκ νεκρῶν ἀναστῶσιν, οὔτε γαμοῦσιν οὔτε γαμίζονται, ἀλλ᾽ εἰσὶν ὡς ἄγγελοι ἐν τοῖς οὐρανοῖς."

This means (in my own translation):

"For, when they rise from the dead, they neither marry nor are given in marriage, but rather they are like the angels in the heavens."

The two later Synoptic Gospels of Matthew and Luke, which were both most likely written sometime between c. 75 and c. 100 CE, reproduce slightly different versions of this line. gMatthew 22:30 portrays Jesus as saying:

"ἐν γὰρ τῇ ἀναστάσει οὔτε γαμοῦσιν οὔτε γαμίζονται, ἀλλ᾽ ὡς ἄγγελοι ἐν τῶ οὐρανῶ εἰσιν."

This means:

"For, in the resurrection, they neither marry nor are given in marriage, but rather are like the angels in heaven."

gLuke 20:36 portrays Jesus as saying:

"οὐδὲ γὰρ ἀποθανεῖν ἔτι δύνανται, ἰσάγγελοι γάρ εἰσιν, καὶ υἱοί εἰσιν θεοῦ, τῆς ἀναστάσεως υἱοὶ ὄντες."

This means:

"For they are no longer able to die, since they are equal to the angels, and they are children of God, being children of the resurrection."

The scholar M. David Litwa in his paper "Equal to Angels: The Early Reception History of the Lukan ἰσάγγελοι (Luke 20:36)" (published in 2021 in the Journal of Biblical Literature, 140(3), 601–622) highlights the fact that gLuke uses a different phrase to describe people in the coming kingdom of God from gMark and gMatthew. Where gMark and gMatthew have the phrase "ὡς ἄγγελοι," which means "like angels," gLuke uses the phrase "ἰσάγγελοι," which means "equal to the angels."

Litwa argues convincingly that the phrase in gLuke conveys a subtly different meaning from the one conveyed in gMark and gMatthew. Where gMark and gMatthew imply that people in the kingdom of God are similar to the angels in the particular respect that they neither marry nor are given in marriage, gLuke seems to imply, more radically, that people in the kingdom of God are beings ontologically equal to angels in general and in all respects.

Regardless of whatever the original authors of these passages intended, early Christian writers in the first few centuries CE already interpreted these passages in the Synoptic Gospels to mean that Christian believers will become beings equal to angels on an ontological level in the afterlife.

Notably, the early Christian philosopher and theologian Clement of Alexandria (lived c. 150 – c. 215 CE) in his Stromateis 4.25 approvingly quotes the Athenian philosopher Plato (lived c. 429 – c. 347 BCE) to say that the souls of believers become like angels rapt in the perpetual contemplation of God after death. He writes, as translated by William Wilson for the Ante-Nicene Fathers series, with some edits of my own to make it more closely match the Greek:

"Rightly, then, Plato says, 'that the man who devotes himself to the contemplation of ideas will live as a god among men; now the mind is the place of ideas, and God is mind.' He says that he who contemplates the unseen God lives as a god among men. And in the Sophist, Socrates calls the stranger of Elea, who was a dialectician, 'god': 'Such are the gods who, like stranger guests, frequent cities. For when the soul, rising above the sphere of generation, is by itself apart, and dwells amidst ideas,' like the koryphaios [lit. "chorus leader"] in Theaitetos, now become as an angel, it will be with Christ, being rapt in contemplation, ever keeping in view the will of God."

You may notice that all the writers I have quoted describe the souls of believers in the kingdom of God or the afterlife as being "like angels" or "equal to angels," but none of them flat-out say that they will be angels. The distinction between someone becoming a being ontologically equal to an angel in the afterlife and them flat-out becoming an angel, however, is very thin and, even in antiquity, it is likely that some Christians believed that they would flat-out become angels after death.