There is no New Testament authority for establishing Sunday as a holy day for believers. There are only eight references to the first day of the week in the NT, and six of those are part of the resurrection narratives. None of the eight can be reasonably construed as establishing the first day of the week as a replacement for the Sabbath.
But whatever the reason, Sunday eventually became the prominent Christian day of worship, displacing the Sabbath from that role. But this change was a gradual one. The earliest clear evidence is found in two 2nd-century documents: the Epistle of Barnabas (from Alexandria, Egypt), and the writings of Justin Martyr (from Rome.)
But the issue was far from settled in the 2nd century.
For hundreds of years the Sabbath and Sunday (“the Lord’s day”) were both observed. The bishops at the council of Laodicea (mid-4th century) agreed on a change in the Sabbath church services: “The Gospels are to be read on the Sabbath, i.e. Saturday, with the other Scriptures.” (Canon 16)
(Remember, we’re talking about what happened roughly 300 years after the crucifixion and resurrection of Jesus.)
“Christians must not Judaize [refrain from labor] by resting on the Sabbath, but must work on that day, rather honouring the Lord’s Day; and, if they can, resting then as Christians. But if any shall be found to be judaizers, let them be anathema from Christ.” (Canon 29)
This shows us that Sabbath observance survived for hundreds of years after the time of Christ. The institutional church, represented by the bishops, wanted to exalt Sunday [the Lord’s day] by dampening the popularity of the Sabbath and the Sabbath rest.
Their efforts were not completely successful, at least outside the church’s two great centers of influence – Rome and Alexandria. Another century and more went by before Socrates Scholasticus wrote:
“For although almost all churches throughout the world celebrate the sacred mysteries on the sabbath of every week, yet the Christians of Alexandria and at Rome, on account of some ancient tradition, have ceased to do this.” (The Ecclesiastical History of Socrates Scholasticas, Book V Chapter XXII)
The biblical day of sacred time continued to be observed. The Sabbath survived.