My answer will try to address the historical perception of a fetus, rather than the criminality of abortion, but there will be some overlap. To maintain neutrality, I'll use the more legal "fetus/foeticide" and "natural person/homicide". I'm also strictly examining historical interpretations of theology, not modern theology, or my own.

I'll mostly address changes in "mainstream" Jewish perception, because I'm not familiar enough with Christian apocrypha.

As a benchmark, by the 6th century, the Talmud(s) had been compiled to form the basis of Rabbinic Judaism. It is not a primary source, but explicitly stated that a fetus, whilst of great value, was not equivalent to the mother's life because it was "part of the mother" and not an individual, animate natural person. It distinguished between the "moment of conception" and the "moment of determination", that up until 40 days a fetus was "like water". It explicitly established a monetary fine as punishment for foeticide.

The Talmud includes a 3rd century source citing a 2nd century source (Rav Meir) saying that only once the fetus crowned, it was considered an "animate being" and its soul of equal value to the mother.

Question answered, right? Well no, the divergence may have occurred earlier.

The main theological issue stems from the interpretation of a specific passage from Exodus:

This passage only refers to miscarriage induced by a deliberate altercation. There are four historical points of contention here:

• The translation of the word ason, as harm/damage/calamity.
• The interpretation of "soul for soul, eye for eye".
• The interpretation of the victim of "other damage".
• The influence of other scriptures on its overall interpretation.

From a literal perspective, there's a separation between foeticide and "other damage". But your question is on the theological perception of a fetus throughout history, not abortion.

The Babylonian Code of Hammurabi (c 1750 BCE) stated something similar, punishing foeticide with a fine, but the death of the mother as a capital crime. This was mirrored by other empires (except the Assyrians, who punished self-inflicted abortion with impalement). This doesn't tell us how they perceived a fetus, only that they considered killing a fetus different to killing a natural person.

This is where it gets complicated. The historicity of the Torah is a bit of a minefield: here's one AskHistorians thread, here's another. In addition, there's no clear consensus on when the Hebrew Bible (Tanakh) canon became fixed. In case you're not Jewish, the Written Torah is only one part of Tanakh, and the Tanakh's interpretation is based on commentaries that vary throughout history (more on this later).

This complicates things further, because translations differ from each other. The Septuagint (Koine Greek translation of the Pentateuch, c. 3rd-2nd century BCE) and some targumim (Aramaic translations of oral works) were written to imply ason applied to the fetus, not the mother.

Why does this matter? Because the Septuagint initially translated ason as form, not damage! This leads to the translation: "if [there is] full form, then you shall give life for life" (ἐὰν δὲ ἐξεικονισμένον ἦν, δώσει ψυχὴν ἀντὶ ψυχῆς). Later Christian scholars corrected this to "harm".

Philo (1st century BCE) and Josephus (1st century CE), as I mention in a comment to another answer, made distinctions between killing an unformed/formed fetus and an infant. Philo relied on the Septuagint for his translations, and Philo and Josephus both follow the Septuagint's translation for this particular passage.

The influence of the Septuagint and Hellenistic Judaism is considerable - even though it differed from the original Hebrew (in some cases considerably) it was used by Jews who couldn't speak Hebrew during the Second Temple period, and by the earliest non-Jewish Christians. It isn't clear if its usage was universal amongst Hellenic Jews. Jews later discarded the Septuagint by the 2nd century CE for Hebrew or Aramaic scriptures. By the 4th century CE, the Septuagint contained inserted Christian apocrypha that weren't present in the original.

Both Philo and Josephus are representative of Judaeo-Hellenic and Judaeo-Roman syncretism - they wrote their texts as assimilated members of Greek and Roman society. Their interpretations therefore are not necessarily representative of mainstream interpretations of Hebrew texts at the time.

It is also why Philo, an Alexandrian Jew, considered abortion a capital crime, but Josephus who would have been raised under the Palestinian school, does not: the phrase "soul for a soul, eye for an eye" for example, is usually interpreted in Hebrew texts as a pecuniary punishment, rather than a literal one. This aligns with the idea that Philo - whose grasp of Hebrew was poor - considered abortion a capital crime (life for life), but Josephus did not.

Remember how the Torah is one part of the Tanakh? Well, the Tanakh is interpreted using the Oral Torah, which was transmitted orally until 70CE. The major compilations of the Oral Torah weren't written down until ~200CE, and weren't finalised until the 6th century. This makes it difficult to pinpoint the exact date of theological divergence. But there are clues.

The first clue is the concept that preserving life overrides all other commandments except homicide (and a few other things). The first explicit Talmudic reference to this cites Samuel of Nehardea (2nd-3rd century), but there are biblical references that predate it.

The other clues are sections of the Oral Torah with detailed explanations on the status of a fetus. In the Tractate Bekhorot, Rav Chisda (3-4th century) is cited as saying it takes 40 days for a fetus to form. The distinction between formed and unformed that Philo and Josephus made is no longer mentioned.

Another Talmudic source stated that before, a fetus is "like water". Likewise, a deceased fetus was said to not impart the same ritual impurity as a person's corpse.

The Mishnah Oholot (3rd century, but quoting a source from ~150CE) allows surgical abortion to save the life of the mother up until crowning. Since homicide was not a permissible reason to save a person's life, a fetus could not have been considered a person.

Assuming (and it is an assumption) that the Mishnah Oholot is correctly citing a century-old source, that narrows it down to before the 2nd century. The question now is what happened before the Mishnah was written down?

Well, that goes back to the Septuagint's translation in the 3rd century BCE. Given the widespread use of the Septuagint, Hellenised Jews who didn't speak Hebrew were usually raised under the Alexandrian school and therefore used different texts to other Jews who followed the Palestinian school.

That's the origin of the divergence, which would have been codified in Christian theology given that non-Hebrew speaking Christians could only have read the Septuagint. I can't say if this divergence was universal within the Alexandrian school.

Regardless, by the final compilation of the Mishnah in the 3rd century and the rest of the Talmud in the 6th century, Rabbinic Judaism did not consider a fetus to be a fully-fledged "animate being", but part of the mother.

By the 11th century onwards, the word ason was considered to apply to the mother rather than the fetus. Later scholars from the 11th century onwards concretised this further by considering a fetus as "not a person", and then by considering a fetus to be an "aggressor" for endangering the life of the mother.

So there you have it. A word, translated differently in the 3rd century BCE, was the basis for the theological divide on the status of a fetus.

EDIT: Spelling. Corrections from mod remarks. Added Koine Greek.

In the Greco-Roman world in the first three centuries CE, under the influence of customary pagan belief and practice, abortion and the abandonment of unwanted infants was commonplace, and not terribly controversial. These two acts were also identified with one another as generally a single practice. There were pagan critics of the practice, notably Cicero in his defense of Aulus Cluentius Habitus. And there is some debate as to whether the practice was more seen as a "regrettable necessity", in the words of McCulloch, or was seen callously as a routine part of life. But in either case, the widespread practice of abortion and infant-killing was a widely-understood fact of life in the pre-Christian Mediterranean.

Jews, however, seem to have rejected this practice from at least the 2nd or 1st century BCE, (if not much earlier), and this discrepancy did not go unnoticed by their Greek neighbors. Tacitus notes (Chapter 5) that they consider killing infants to be a crime. He does not consider this to be a compliment, and lists this custom among others he calls "base and abominable". Whether or not he is correct in his assertion that Jews categorically considered it unlawful, it certainly speaks volumes that he considered it a barbaric trait for them to do so.

As to if and when abortion, as a procedure performed prior to natural birth, became divested from this broader concept in Jewish law, I will leave to another commenter, but from my research and understanding of Mediterranean late-antiquity, it was not yet so distinguished.

Christianity, in the first several centuries of its existence, as you're probably aware, experienced an acute identity crisis between its Hebrew and Greek cultural heritage. Many early Christian writers, such as Tertullian, the author of the Epistle to Barnabas, and the author(s) of the Didache, (to name a few), all embraced their Greek inheritance. They insisted that one could be a Hellene and a Christian, and indeed that the titans of Hellenic Philosophy comported with Christian teachings. However, despite the fact that these authors were happy to underline their Hellenic character, and did not at all seek to emulate Jewish culture and practice wholesale, even these Greek-leaning Christians broke radically with their pagan kinsmen on the question of abortion, and accepted the Jewish custom of considering it to be both a sin and a crime.

All early Christian communities abhorred abortion and infant-killing, and Jewish sentiment too was consistently opposed to it - Josephus and Philo both expressly forbade it. The Didache, the Apocalypse of Peter, and the Apolegoticum, are examples of early Christian writings that also expressly forbade it.

In short, for the first several centuries of their co-existence, Judaism and Christianity were in accord. They considered infant-killing to be a crime, and they considered invasive abortion prior to birth as the same thing. Polemics between Jews, Jewish Christians, and Hellenic Christians tend to obscure areas of alignment, but it's clear that from the perspective of the outsider - the pagan Greco-Roman - Christians and Jews shared the same position at least until the fourth century.

For more information on how early Christian communities distinguished themselves from Hellenic society, and in what ways they accommodated themselves, see The self-defining praxis of the developing ecclsesia by Carolyn Ostek, published as Chapter 14 of The Cambridge History of Christianity, edited by Mitchell and Young.

For more information on the practice of infant killing, infant abandonment, and abortion in antiquity (and beyond), see The Kindness of Strangers by John Boswell. Excellent work.

I also consulted Christianity - The First Three Thousand Years by Diarmaid McColluch, specifically Chapter 4, "Boundaries Defined".