How closely after Christ's ascension in 30 CE would count? What about before Christ's birth and death and resurrection and ascension?
There are of course many prophecies about Messiah in the Old Testament (OT) and as many commentators about those prophecies. The OT was written 400 plus years before Christ. For example, it speaks to where and how He would be born. It provides a precise date line. It tells us facts about Messiah's death, burial, and resurrection. It should thus be no surprise that what we have today is of a particularly necessary consequence to those fulfilled prophecies that were met in Christ Jesus. Indeed we have numerous accounts from Christians within 150 years of His ascension. There's Polycarp, Papias, Melito, Justin Martyr, the Didache, Clement of Rome, and others. Not to mention of course the New Testament.
Here for example is a snippet from Tertullian circa CE 200 who will say this about that. He admits our writings date back no further than to Tiberius who died in the Spring of 37. The apostles went forth via voice and only later wrote down the Truth. Others then did the same.
But having asserted that our [Christian] religion is supported by the writings of the Jews, the oldest which exist, though it is generally known, and we fully admit that it dates from a comparatively recent period—no further back indeed than the reign of Tiberius—a question may perhaps be raised on this ground about its standing, as if it were hiding something of its presumption under shadow of an illustrious religion, one which has at any rate undoubted allowance of the law, or because, apart from the question of age, we neither accord with the Jews in their peculiarities in regard to food, nor in their sacred days, nor even in their well-known bodily sign, nor in the possession of a common name, which surely behoved to be the case if we did homage to the same God as they. Then, too, the common people have now some knowledge of Christ, and think of Him as but a man, one indeed such as the Jews condemned, so that some may naturally enough have taken up the idea that we are worshippers of a mere human being. But we are neither ashamed of Christ—for we rejoice to be counted His disciples, and in His name to suffer—nor do we differ from the Jews concerning God. We must make, therefore, a remark or two as to Christ’s divinity.
But what of non-Christian sources? What about those who don't particularly care about the truth of who Christ is, but rather, report the truth that Christ existed? We have three very early, non-biased sources who tell us that Christ existed. Josephus wrote about 95. Tacitus wrote about 115. Pliny the Younger wrote also about these times. They all know of Christ and Christians. More importantly their history aligns to the gospel accounts. Again, they don't believe the truth, but they report the fact there is a truth.
The more important quote from Josephus is really this where he allows the title of Jesus Christ, while not informing that is his belief, which it wasn't. Some don't like this passage because it implies Mary and Joseph consummated their marriage with James as one of the offspring. In other words, to agree with this earliest non Christian account is to admit something about Mary and Joseph. 'Course this too aligns to the gospel accounts about Jesus' brothers.
Festus was now dead, and Albinus was but upon the road; so he assembled the sanhedrim of judges, and brought before them the brother [James] of Jesus, who was called Christ,
Consequently, to get rid of the report, Nero fastened the guilt and inflicted the most exquisite tortures on a class hated for their abominations, called Christians by the populace. Christus, from whom the name had its origin, suffered the extreme penalty during the reign of Tiberius at the hands of one of our procurators, Pontius Pilatus, and a most mischievous superstition, thus checked for the moment, again broke out not only in Judæa, the first source of the evil, but even in Rome, where all things hideous and shameful from every part of the world find their centre and become popular. Accordingly, an arrest was first made of all who pleaded guilty; then, upon their information, an immense multitude was convicted, not so much of the crime of firing the city, as of hatred against mankind.
PLINY the YOUNGER
He writes to the Emperor Trajan about 112 CE regarding Christ and Christians.
They asserted, however, that the sum and substance of their fault or error had been that they were accustomed to meet on a fixed day before dawn and sing responsively a hymn to Christ as to a god, and to bind themselves by oath, not to some crime, but not to commit fraud, theft, or adultery, not falsify their trust, nor to refuse to return a trust when called upon to do so.
So no, these accounts were not written in the years CE 25-35. But they were written within one generation thereto. I could again mention Polycarp who wrote about 150 and who was taught directly by John the Apostle son of Zebedee who was taught by Jesus Christ, but some may think Polycarp biased in his letter. The three Josephus, Tacitus, and Pliny the Younger, however, have no dog in the hunt to prove or disprove the truth that a man some deemed the prophesied Messiah walked the earth. They simply tell us that Christ Jesus did exist and did have followers. To believe the truth is up to you.