This is all I could glean from "scholarly sources" about this matter. First, under the heading "Biblical Concept of Salvation", this Catholic source shows a change in view after the destruction of Israel and Judah; it was viewed as bringing home the 'remnant', which would result in a kingdom of peace in which God reigns as king.
"In the post-exilic period, there appears as well as God the figure of
an actual bringer of salvation; cf. the prince of peace, Zech 9:9.
...later books of the OT show the development of the idea that on the
day of judgment Israel can expect final salvation but the (pagan)
nations which have oppressed Israel must expect final perdition (Wis
5:2...) This restriction of the idea of salvation to Israel appears
even more strongly in the non-biblical books of Judaism, e.g.
Jubilees, Psalms of Solomon, Enoch. They hold in common that the
Gentiles were really created only for destruction...
In later Judaism the Torah was regarded as a saving gift because
with its help men could faithfully fulfil the commandments and thus
acquire merit for themselves. God must pay them a well-earned reward
in the next world...
In the NT the word salvation is a religious term... any healing is a
sign of the bestowal of salvation by Jesus. With Jesus, salvation has
come to men. ... Man of himself can effect no salvation; even faith,
conversion, baptism and constancy in earthly life acquire for him no
'right' to salvation, but are only its necessary presuppositions.
Salvation is not restricted to particular groups, as in the OT and in
Qumran, but extends in principle to all men because of the universal
efficacy of Jesus' death." Encyclopedia of Theology, pp.1504-6,
article by Ingrid Maisch, Burns & Oates, 1981
Second, under the heading "The Christian church and the Jews" another source makes this point:
"At first Christians were regarded as a Jewish sect by both Jews and
Gentiles. This led to opposition and persecution of the church by the
Jewish authorities, who objected to its doctrines and the admission of
Gentiles without their accepting the Law." The History of
Christianity, p.50, this article by H. L. Ellison, Lecturer & writer
on the OT, Dawlish, England, Lion, 1977
This is the matter raised in Acts 15:1. It is about Gentiles becoming Christians without having been circumcised, as were all the Jewish Christians, up to that point. They were circumcised on the eighth day from birth, but Gentile converts never had been. There was a move to get them circumcised, but this is what the apostles showed to be contrary to the Christian faith. Salvation was based on faith in Jesus Christ, but anyone getting circumcised obliged themselves to keep all of the Mosaic Law. The writings of the early Church (as detailed in the book of Acts) show that not only was circumcision unnecessary, it would actually violate the Christian stance that salvation from God to individuals came with their putting total faith only in what Christ had done to save them from their sin.
The Jewish take on salvation was very different. The faith of Abraham and other faithful people of old was lauded, but their hope (as a nation) was for the Messiah to come, to remove the hated Roman yoke (at that time) and re-establish a literal throne of David. They thought about being saved from their enemies. It was only with the advent of Jesus Christ that the concept of being saved from sin by God, through faith in Christ, emerged. After his resurrection, the new Church burst onto the scene, first with Jewish converts, then with Gentiles being added en masse.
In the chapter entitled 'What the First Christians Believed', it says more about Christianity and the Jews, including some of what I have stated above, so I won't repeat that as a quote. But here are relevant points:
"The resurrection of Jesus was emphasized more than his death in the
earliest preaching to Jews, because it demonstrated that the man
executed as a criminal was nevertheless God's Messiah. Following
guidelines laid down by Jesus himself, the apostles pointed to Old
Testament passages which had been fulfilled in his career and in the
beginnings of the church. 'This is what was prophesied' was a phrase
frequently on their lips...
But all early Christian theology was Jewish, since the language and
concepts it used were quarried chiefly from the Old Testament. Some
Jewish Christians were so conservative that they demanded, in effect,
that Gentiles had to become Jews in order to be true Christians. They
insisted on circumcision and other Jewish legal requirements, and
frowned on social contact with 'unclean' Gentiles. These 'Judaizers'
appealed to the Jerusalem church where James led a community of
thousands of 'staunch upholders of the Law'...
In Jerusalem the harmony maintained between James and the Jewish
authorities failed to survive his martyrdom in AD 62, and the Jewish
war with Rome which began four years later. Jewish-Christian
relations continued to deteriorate later in the first century. Judaism
entrenched itself within the tight limits set by the rabbinic
Pharisees. It excluded non-conformists like the followers of Jesus.
Conservative Jewish-Christianity disappeared into obscurity. Its
strength filtered off into side-channels, such as the heretical
Ebionite groups. It may also have merged with currents from other
brands of Judaism... Fringe Judaism of one kind or another fertilized
the emerging Gnostic sects which loomed so large in the second
century." Ibid. pp. 97-100,
article by David F. Wright, Snr. Lecturer in Ecclesiastical History,
Uni. of Edinburgh Scotland
He also pointed out that the martyr Stephen boldly declared the old covenant obsolete, and that others in the next century portrayed Israel as an unbelieving and apostate people, which was why so few Jews in total responded to Christ's message of salvation from sin. Christians find their identity in Christ, not in belonging to any nation - they now belong to the Church, the Bride of Christ, which is spiritual Israel.
I have looked at other books but there is nothing about a definition of salvation given by the earliest Church outside of what they wrote in the New Testament. Most scholarly books seem to launch out from the second century onward, which is not the era you are looking at. Before Saul of Tarsus was converted and became Paul, the apostles spoke and wrote about salvation and life eternal commencing with repenting of sin and trusting only in Jesus, "to all who will believe". The book of Acts is full of that, even up to ch. 15, before Paul came along. Peter's address on the Day of Pentecost is about salvation from sin, and three thousand Jews and proselytes were converted on that day.
Jews thought in terms of salvation as God's chosen people, the nation of Israel, obliged to keep all the laws to please God, to show the world that they were a holy nation, exemplifying the worship and protection of God as a nation. Christianity opened up salvation to Gentiles without them having to keep that law in order to be saved. Salvation was for the Church, which included Jews and Gentiles, a spiritual, global people of God. But for the Jews salvation was for their nation, and Gentiles could only enter into that by being circumcised and keeping the law of Moses. For the Jews it was a system of law-keeping. For the Christians it was God bringing them into relationship through Christ, sin being the barrier, but faith brought them into the family of God, which now had nothing to do with physical nationality. That shows the crucial differences between Jewish views of salvation, and Christian ones, during the first century A.D.