Just wondering in what year the first historical record of the Church in Rome, against the wishes of other churches, that she should impose her tradition on them?
Note: I am simply looked for the first factual recorded date in history. I am not making an inquiry into whether the church in Rome had a right, or not, to do so.
I'm frankly dumbfounded. Unless I'm reading wrongly, @Mike has asked this question, and then provided his own answer.
To answer your question, though it hardly seems necessary since you knew your answer before asking your question, before "the church in the city of Rome" tried "to impose a tradition on other churches", whatever that means, other churches were querying the church of Rome for guidance. Pope St. Clement (+AD 96) wrote to the church of Corinth in response to complaints from its leadership. The presbyters and bishop of Corinth had been deposed and chased out of the city and they wrote to Rome asking its bishop to validate their authority as appointees of the apostles. And this happened perhaps while St. John was still even alive.
It is wise, when wondering how Rome started "imposing traditions other churches," to remember that while Rome had armies, the church of Rome had none. How, precisely, is the Bishop of Roman supposed to have imposed anything on anyone?
The appeal to Rome is always an appeal to moral authority. Even in the heyday of her power, the Church of Rome and the Papal States were by-and-large reliant on other nations' for their military protection. When patriarchs in Constantinople and bishops in North Africa ask for the support of the Pope, they are not asking for his armies - at his most militarized he had precious few - they are asking for his endorsement.
The authority of Rome may have been presumed because of the time St Paul spent there. He died AD 64/5 and probably some of the Corinthians who heard him teach were still alive when they wrote to Rome.
At least four of Paul's doctrinal letters were written from Rome and it seems to be where his days ended, though there is some doubt. Many scholars agree that St Paul's letters were the first to be recognized as inspired by the early church, and certainly no other writings rise to the level of his prison epistles.
There seems to have been a flourishing Christian community in Rome (see Romans 16 and Philippians) and possibly the Corinthians thought some of Paul's adherents would give an answer to their questions.
The earliest record I have been able to find was between 189-199 AD. This occurred when a dispute over what day the celebration of Easter should occur. This happened betwwen the Christians at Rome and those from the province of Asia. Victor, the bishop of Rome decided to act dictatorial actually trying to excomminicate whole churches over an extremely non essential matter, naturally greatly offending other bishops.
Victor called a meeting of Italian bishops at Rome, which is the earliest Roman synod known. He also wrote to the leading bishops of the various districts, urging them to call together the bishops of their sections of the country and to take counsel with them on the question of the Easter festival. Letters came from all sides: from the synod in Palestine, at which Theophilus of Caesarea and Narcissus of Jerusalem presided; from the synod of Pontus over which Palmas as the oldest presided; from the communities in Gaul whose bishop of Irenaeus of Lyons; from the bishops of the Kingdom of Osrhoene; also from individual bishops, as Bakchylus of Corinth. These letters all unanimously reported that Easter was observed on Sunday. Victor, who acted throughout the entire matter as the head of Catholic Christendom, now called upon the bishops of the province of Asia to abandon their custom and to accept the universally prevailing practice of always celebrating Easter on Sunday. In case they would not do this he declared they would be excluded from the fellowship of the Church.
This severe procedure did not please all the bishops. Irenaeus of Lyons and others wrote to Pope Victor; they blamed his severity, urged him to maintain peace and unity with the bishops of Asia, and to entertain affectionate feelings toward them. Irenaeus reminded him that his predecessors had indeed always maintained the Sunday observance of Easter, as was right, but had not broken off friendly relations and communion with bishops because they followed another custom. (Catholic Encyclopedia as posted here)
So it seems around 90-100 years after the last Apostle died, did the the first historical record appear when church of Rome tried to impose its tradition on other churches who resisted that attempt. This was around the year 195 AD.
In the time of this Clement, no small dissension having occurred among the brethren at Corinth, the Church in Rome dispatched a most powerful letter to the Corinthians, exhorting them to peace, renewing their faith, and declaring the tradition which it had lately received from the apostles, proclaiming the one God, omnipotent, the Maker of heaven and earth, the Creator of man, who brought on the deluge, and called Abraham, who led the people from the land of Egypt, spoke with Moses, set forth the law, sent the prophets, and who has prepared fire for the devil and his angels.