@Ken Good point about 1 Corinthians 15. But it is interesting how distinctive that passage is in relation to the hundreds of mentions Christ in the Paulines.
As for Roman Christians understanding Jewish messianism, I would think that the events of the Second Jewish-Roman War would have made this pretty clear.
From First Apology:
They are also in the possession of all Jews throughout the world; but they, though they read, do not understand what is said, but count us foes and enemies; and, like yourselves, they kill and punish us whenever they have the power, as you can well believe. For in the Jewish war which lately raged, Barchochebas, the leader of the revolt of the Jews, gave orders that Christians alone should be led to cruel punishments, unless they would deny Jesus Christ and utter blasphemy. In these books, then, of the prophets we found Jesus our Christ foretold as coming, born of a virgin, growing up to man's estate, and healing every disease and every sickness, and raising the dead, and being hated, and unrecognised, and crucified, and dying, and rising again, and ascending into heaven, and being, and being called, the Son of God. We find it also predicted that certain persons should be sent by Him into every nation to publish these things, and that rather among the Gentiles [than among the Jews] men should believe in Him. And He was predicted before He appeared, first 5000 years before, and again 3000, then 2000, then 1000, and yet again 800; for in the succession of generations prophets after prophets arose.
Moses then, who was the first of the prophets, spoke in these very words: The sceptre shall not depart from Judah, nor a lawgiver from between his feet, until He come for whom it is reserved; and He shall be the desire of the nations, binding His foal to the vine, washing His robe in the blood of the grape. Genesis 49:10 It is yours to make accurate inquiry, and ascertain up to whose time the Jews had a lawgiver and king of their own. Up to the time of Jesus Christ, who taught us, and interpreted the prophecies which were not yet understood, [they had a lawgiver] as was foretold by the holy and divine Spirit of prophecy through Moses, that a ruler would not fail the Jews until He should come for whom the kingdom was reserved (for Judah was the forefather of the Jews, from whom also they have their name of Jews); and after He (i.e., Christ) appeared, you began to rule the Jews, and gained possession of all their territory. And the prophecy, He shall be the expectation of the nations, signified that there would be some of all nations who should look for Him to come again. And this indeed you can see for yourselves, and be convinced of by fact. For of all races of men there are some who look for Him who was crucified in Judæa, and after whose crucifixion the land was straightway surrendered to you as spoil of war. And the prophecy, binding His foal to the vine, and washing His robe in the blood of the grape, was a significant symbol of the things that were to happen to Christ, and of what He was to do. For the foal of an ass stood bound to a vine at the entrance of a village, and He ordered His acquaintances to bring it to Him then; and when it was brought, He mounted and sat upon it, and entered Jerusalem, where was the vast temple of the Jews which was afterwards destroyed by you. And after this He was crucified, that the rest of the prophecy might be fulfilled. For this washing His robe in the blood of the grape was predictive of the passion He was to endure, cleansing by His blood those who believe in Him. For what is called by the Divine Spirit through the prophet His robe, are those men who believe in Him in whom abides the seed of God, the Word. And what is spoken of as the blood of the grape, signifies that He who should appear would have blood, though not of the seed of man, but of the power of God. And the first power after God the Father and Lord of all is the Word, who is also the Son; and of Him we will, in what follows, relate how He took flesh and became man. For as man did not make the blood of the vine, but God, so it was hereby intimated that the blood should not be of human seed, but of divine power, as we have said above. And Isaiah, another prophet, foretelling the same things in other words, spoke thus: A star shall rise out of Jacob, and a flower shall spring from the root of Jesse; and His arm shall the nations trust. Isaiah 11:1 And a star of light has arisen, and a flower has sprung from the root of Jesse — this Christ. For by the power of God He was conceived by a virgin of the seed of Jacob, who was the father of Judah, who, as we have shown, was the father of the Jews; and Jesse was His forefather according to the oracle, and He was the son of Jacob and Judah according to lineal descent.
For the Jews having the prophecies, and being always in expectation of the Christ to come, did not recognize Him; and not only so, but even treated Him shamefully. But the Gentiles, who had never heard anything about Christ, until the apostles set out from Jerusalem and preached concerning Him, and gave them the prophecies, were filled with joy and faith, and cast away their idols, and dedicated themselves to the Unbegotten God through Christ.
From Dialogue with Trypho:
For thus, so far as you are concerned, I shall be found in all respects innocent, if I strive earnestly to persuade you by bringing forward demonstrations. But if you remain hard-hearted, or weak in [forming] a resolution, on account of death, which is the lot of the Christians, and are unwilling to assent to the truth, you shall appear as the authors of your own [evils]. And you deceive yourselves while you fancy that, because you are the seed of Abraham after the flesh, therefore you shall fully inherit the good things announced to be bestowed by God through Christ. For no one, not even of them, has anything to look for, but only those who in mind are assimilated to the faith of Abraham, and who have recognised all the mysteries: for I say, that some injunctions were laid on you in reference to the worship of God and practice of righteousness; but some injunctions and acts were likewise mentioned in reference to the mystery of Christ, on account of the hardness of your people's hearts. And that this is so, God makes known in Ezekiel, [when] He said concerning it: 'If Noah and Jacob and Daniel should beg either sons or daughters, the request would not be granted them.' Ezekiel 14:20 And in Isaiah, of the very same matter He spoke thus: 'The Lord God said, they shall both go forth and look on the members [of the bodies] of the men that have transgressed. For their worm shall not die, and their fire shall not be quenched, and they shall be a gazing-stock to all flesh.' Isaiah 66:24 So that it becomes you to eradicate this hope from your souls, and hasten to know in what way forgiveness of sins, and a hope of inheriting the promised good things, shall be yours. But there is no other [way] than this— to become acquainted with this Christ, to be washed in the fountain spoken of by Isaiah for the remission of sins; and for the rest, to live sinless lives.
For we all expect that Christ will be a man [born] of men, and that Elijah when he comes will anoint him. But if this man appear to be Christ, he must certainly be known as man [born] of men; but from the circumstance that Elijah has not yet come, I infer that this man is not He [the Christ].
This goes on for some time talking about proofs that Jesus is the Christ the Jews were expecting.
Trypho: All the words of the prophecy you repeat, sir, are ambiguous, and have no force in proving what you wish to prove.
Justin: If the prophets had not ceased, so that there were no more in your nation, Trypho, after this John, it is evident that what I say in reference to Jesus Christ might be regarded perhaps as ambiguous. But if John came first calling on men to repent, and Christ, while [John] still sat by the river Jordan, having come, put an end to his prophesying and baptizing, and preached also Himself, saying that the kingdom of heaven is at hand, and that He must suffer many things from the Scribes and Pharisees, and be crucified, and on the third day rise again, and would appear again in Jerusalem, and would again eat and drink with His disciples; and foretold that in the interval between His [first and second] advent, as I previously said, priests and false prophets would arise in His name, which things do actually appear; then how can they be ambiguous, when you may be persuaded by the facts? Moreover, He referred to the fact that there would be no longer in your nation any prophet, and to the fact that men recognised how that the New Testament, which God formerly announced [His intention of] promulgating, was then present, i.e., Christ Himself; and in the following terms: 'The law and the prophets were until John the Baptist; from that time the kingdom of heaven suffers violence, and the violent take it by force. And if you can receive it, he is Elijah, who was to come. He that has ears to hear, let him hear.'
Actually much of Trypho dals with how Jesus fulfilled prophecies which the Jews misunderstand and have misapplied, it just goes on and on.
As for the Romans, the Romans were deeply obsessed with prophecy. Their culture was more obsessed with prophecy than just about any culture in history. Their whole government was organized around prophecy, and prophecy was central to the administration of the Senate. In fact every Senator was a "prophet", an augur who was instructed in the auspices.
A common belief of Greeks, Romans, and many other Mediterranean cultures, was that unfulfilled prophecies may yet come true, but that once a prophecy had been fulfilled then that was the end of it. Unfulfilled prophecies were often seen as a looming danger, but once it was identified that a prophecy had been fulfilled then there was relief, as of course many prophecies were prophecies of doom.
Are there more explicit statements of Romans in the 2nd or 3rd century saying that the Jews should stop looking for their messiah because he has already come? I'm not sure. I haven't explicitly gone looking for these, but I would certainly be interested to know if there are.