What is the earliest historical testimony (e.g., in writing) of the celebration (i.e., a feast) of the nativity of the Lord Jesus Christ, whether it be on the equivalent of December 25, January 6, or some other date of the Roman calendar?
Attitudes on birthdays
In the early church, birthdays (in general) were not seen as something to celebrate. For example, in Origen's 8th homily on Leviticus he writes:
But the saints not only do not celebrate a festival on their birth days, but, filled with the Holy Spirit, they curse that day.
According to an article by Andrew McGowan (Bible Review, December 2002; republished by the Biblical Archaeology Society), this is "a strong indication that Jesus’ birth was not marked with similar festivities at that place and time [early third century]."
Interest in the date of Jesus' birth & Epiphany
There was, however, some interest in the date of Jesus' birth, even though it was not yet a feast date. Around the year 200, Clement of Alexander wrote
And there are those who have determined not only the year of our Lord's birth, but also the day (Stromata Book I, Chapter 21)
Clement then offers a couple different calculations as to the date - the 25th of Pachon [May 20] or the 24th or 25th of Pharmuthi [April 20 or 21]. Perhaps more notably for purposes of this answer, he also writes:
And the followers of Basilides hold the day of his baptism as a festival, spending the night before in readings. And they say that it was the fifteenth year of Tiberius Cæsar, the fifteenth day of the month Tubi [January 10]; and some that it was the eleventh of the same month [January 6].
The Basilidians were a Gnostic group that taught Jesus became God at His baptism. Their celebration of the Epiphany on January 6/10, which was originally focused on Jesus' baptism in mainstream Christianity as well, provides the first evidence for the feast. In Eternity Today (Bloomsbury Publishing, 2006), Martin Connell explains how this evidence coincides well with what is known about the rise of Christianity in Egypt.
Shortly after Clement's Stromata, Hippolytus wrote his Commentary on Daniel. The work, although not extant in the original Greek, has been recovered from a combination of Patristic quotes (primarily catenas - collected verse-by-verse commentaries) and a Syriac translation. In one passage he writes:
For the first advent of our Lord in the flesh, when he was born in Bethlehem, eight days before the kalends of January [=December 25th], the 4th day of the week, while Augustus was in his forty-second year, but from Adam five thousand and five hundred years.
The "eight days ... the week" clause has generally been thought to be an interpolation. However, Tom Schmidt has recently provided detailed analysis that suggests the date is original. His peer reviewed paper "Calculating December 25 as the Birth of Jesus in Hippolytus’ Canon and Chronicon" (Vigiliae Christianae, Volume 69, Issue 5, October 2015) provides further evidence that Hippolytus' writings support a December 25 date for Jesus' birth.
Origin of the Christmas feast
According to McGowan, the earliest celebrations of Christmas most likely occurred sometime between 250 and 300. Evidence for this comes from the Donatist Christians. In 411, Augustine of Hippo chides the Donatists for refusing to acknowledge the feast of Epiphany, but does not similarly chide them for not celebrating Christmas. This implies it was not a point of dispute between the mainstream Christians and the Donatists, suggesting the Donatists also celebrated the feast. Since this group came into existence no later than 312 and were known to strictly adhere to the then-existent customs, this evidence suggests the Christmas feast likely predates the group.
The first explicit mention of a Christmas feast celebrated on December 25 comes from Roman almanac, dated to 354, that records a Christmas feast on December 25, 336. (Dan Graves, "The 1st Recorded Celebration of Christmas" at Christianity.com)
Although it is a popularly asserted that Christmas was borrowed from pagan culture, that idea does not coincide well with what we know about third century Christianity. (Many of the modern customs of Christmas were, no doubt, borrowed from the culture at various points, but the date itself does not appear to have been borrowed.) Instead, the date for Christmas likely stems from the date for Jesus' crucifixion, as strange as that may seem. This is due to an early tradition that Jesus was conceived on the say day he was crucified, which ultimately stems from a combination Jewish tradition about Biblical figures living exactly the number of years stated in the Bible, combined with philosophical ideas about completeness.
Jesus' birth date, then, was set nine months after His crucifixion/conception date. In the west, where it was thought Jesus was crucified on Nissan 14 (March 25), Christmas became December 25. In the east, they used the 14th day of the Greek calendar instead of the Hebrew, putting the dates at April 6 and January 6 respectively. (See McGowan's article and the numerous sources he cites for additional information.)
If you are looking for the first unambiguous evidence of a Christmas feast, it comes from a 354 almanac documenting the feast back to 336. If you are looking for what the evidence says about the actual origin of the feast, the answer is late third century. If evidence for the Epiphany feast counts, the date can be pushed back to back to the year 200, although that feast was originally primarily about Jesus' baptism (philosophical ideas of the time would say He was most likely baptized exactly 30 years after His birth, so there is some connection regardless). The earliest evidence for interest in determining Jesus' birthday also dates to around the year 200.