The answer is: much later than we might think!
The early church seems to have avoided any titles for Christians, except for the egalitarian “brother” and “sister.” Matthew’s gospel, which is very concerned about the rules of conduct within a Christian community, records this teaching of Jesus: “Call no one on earth your father, you have but one Father in heaven” (Matthew 23:9). Jesus seems to be suggesting that titles are a way of claiming rank over and above others and therefore were not proper for a disciple who sought to be a servant to all.
With the passage of time, however, the title “Father” crept into Christian etiquette as a way of describing the relationship of a priest or monk with those to whom he ministered. The custom arose among the followers of the monastic orders of St. Benedict to address the leader of a monastery as “abbot” and the leader of a convent as “abbess.” This word was a variation of the Aramaic word for father, “abba.”
Bishops also began claiming the title “Father” to describe the nature of their teaching authority over a local church. In the Middle Ages, as the practice of making a private confession to a priest grew, priests who served as confessors were called “Father.” So were mendicant friars like the Franciscans and Dominicans.
Despite these precedents, the use of “Father” as the normal title for ALL priests, whether attached to a diocese or members of a religious order, is a very recent practice. It originated in Ireland and spread to the United States with the Irish immigrations of the 1840’s. When Cardinal Manning was archbishop of Westminster (1865-1892) he worked hard to establish this custom as the universal practice in England as well. It is still largely restricted to English speaking countries (although in France, Catholics use “Pere” in addressing a priest).