That's right, Nadir. It did indeed happen immediately. This is not really necessary to add as your post sums it up, but I believe there is also something the Church Fathers reflect on which is that Sunday is also the first day of creation. The day when God said, "be light made".
Christ's Birth was also thought to occur on a Sunday, and according to Guuaranger, when Christmas falls on a Sunday it is considered to reflect the fullness of that great Feast within the Liturgical Year.
Having said that, I do kind of resent that this is even a question. I am not saying I am troubled by the OP. No, I am resentful that this even has to be brought up for debate due to certain weirdo Christian cults like the Seventh Day Adventists and other assorted Sola Scriptura types who put the Old Testament at a higher place than the New Testament. A lot of this happened because they recognize they have no authority and was meant to challenge Church authority as the world was beginning to embrace a type of Judiazation. These are the same kinds of people who will tell you that the Church got co-opted by Roman politicians and that is where it all went wrong.
This, according to them of course, was followed by 1100 years of "darkness" until the so called Reformation. But even Luther and Henry dared not challenge Sunday as the Lord's Day. But this is the nature of revolution. Eventually other sects have to find differences to justify things like the Second Community Church of the Last Brethren or other such nonsense. Coincidently, that approximate 1,000 year time period was the Golden time for the Church and the time period when the Jєωs were most under control.