It seems this must have been somewhere fairly significantly after the founding and spread of Christianity.
Assuming this was some sort of pre-schism Catholic action, why did the Orthodox and protestant offshoots not revert back to the original schedule?
What peaked my curiosity is linguistics - I'm currently studying Greek. The Greek/Italian/Spanish words for "Saturday" are "Sabado", which is a mangled version of the Hebrew word "Sabbath". From this it seems logical that this change would have had to take place after say the founding of the Byzantine empire, or the spread of Christianity through Rome (eg, the people/leaders would have had to be christian already in order to rename their days of the week to use "Sabbath" - whereas the latin (and English) world for Saturday is obviously pagan from Saturn).
I think a little clarification is helpful here.
Constantine I did declare Sunday a day of rest in the Empire, at least in the cities, but only by way of commending it; in the military, he required Christian soldiers to 'rest' and to attend worship.
But this doesn't amount to a 'change of Sabbath', and this is where clarity is needed. Christians began worshipping on Sunday very early. Revelation 1:10 is usually taken as an indication that by the late 1st century it was the day for Christians to gather. This is (a) not a re-location of the Sabbath, but is (b) aligned with the resurrection.
When Constantine moves to make it 'official', he makes no reference to the Sabbath, or to the 4th commandment and "day of rest", rather it is about establishing Sunday as a common day of worship and freeing Christians to observe and attend worship.
Constantine's successors generally pursued legal policies that reinforced the importance and legal status of Sunday, but this again was not a 'change' of what the Sabbath was.
The idea that Sunday is a new Sabbath thus does not appear in antiquity per se. To the best of my knowledge, that equation does not appear until much later. In English speaking contexts, it is really the Puritans who closely linked the Sunday with the 4th commandment and theologised it as a new, Christian sabbath.
For a tangent, and since you mention your linguistic interest, this eventually leads to the linguistic division in Scottish Gaelic where Sunday has traditionally been "Latha na Sabaid" (Day of the Sabbath) among the staunchly calvinist, and "DiDòmhnaich" (Day of the Lord) among catholics.
To head back to your original question, I think you are making some assumptions about why Greek/Italian/Spanish call Saturday 'Sabado' that do not necessarily hold. A more likely linguistic hypothesis is simply that the continued presence of Jews around the mediterranean basin lead to those languages adopting 'Sabado' for 'Sabbath' and replacing 'Saturday' at the level of vernacular usage, not as an intentional "renaming" of days by leaders (top-down language policy is fraught with difficulty).
On the legislative enshrinement of Sundays in antiquity, Schaff's "From Constantine the Great to Gregory the Great, A.D. 311-600", pages 378-383 gives an overview of the main developments as well as some sources. It's old, but on this topic it's still relatively good.