When Tatian (110 - 180) created the Diatessaron, it appears that he relied exclusively on the four gospels with which we are familiar, lending weight to the idea that some portion of the church recognized all and only those four gospels before 200.
The Muratorion list was described earlier.
Irenaeus, who died around 200, referred to the four gospels, the Acts of the Apostles, and the thirteen lettters of Paul. He also included the Shepherd of Hermas, a book that was later excluded.
Tertullian, during the ten years (197 - 207) he was with what became the mainstream church, provided lists that included the four gospels, Acts, Revelation, I John, I Peter, and Jude. He mentions the letter to the Hebrews as respected but not of the same calibre as the other works. Early in his writings he champions Hermas, but later set that aside.
Clement of Alexandria, (died ca. 215), includes the four Gospels, Acts, the letters of Paul and Hebrews, and Revelation. He also discusses, although it is unclear that he considered all of them scripture, I Peter, I John, Jude, a letter of Barnabas, the Revelation of Peter, the Gospel of the Hebrews, and the Gospel of the Egyptians.
So, before 220, we have testimony of several lists of books, several of which were not later accepted as scripture, but the core of what became the New Testament is already being organized over 100 years prior to Nicaea.
By 250, Origen was already producing a list of what he deemed others in the church had accepted. He noted that different works were treated as scripture in different places, but compiled a list that included: 1) the four gospels, Acts, the thirteen letters of Paul, I peter, I John, and Revelation as held to be scripture by everyone; 2) II Peter, II John, III John, Hebrews, James, Jude, Barnabas, and held by many, but doubted by some (with Hermas and the Didache revered but not part of his canon; and 3) the gopels of Thomas, Matthias, Basilides, and the Egyptians being not recognized as scripture.
Other lists compiled in the same period show different entries, but it is clear that a core (if not-yet-official) canon was already being hammered out well in advance of the Council of Rome (since we can ignore irrelevant comments about Nicaea).