So, it's a good question and it's a little complicated but not too much.

Firstly, there are small indications that Christians in the New Testament period (the mid to late 1st century AD), met on the 'first day of the week'. This can be seen in Acts 20:7 and 1 Cor 16:2, for instance. This reflected the belief that Jesus rose from the dead on the first day, i.e. Sunday (because Saturday is the Sabbath and it's the 7th day). You see this day referred to in Revelation 1:10 as 'the Lord's Day', which became a common designation for it.

Christians appear to have met regularly on the Sunday then, from early on. In Jerusalem, there's some debate about whether Jewish-background believers continued to observe temple gatherings and practices or not, but if they did, then meeting separately on the Sunday as messianic followers would also make sense.

Early 2nd century, you have Pliny the Younger's Letter to Trajan (Epistulae X.96), he makes mention that Christians gather on a certain day of the week, before dawn, to worship. There's no reason to think this isn't Sunday. Justin Martyr, in his First Apology, ch 67, makes clear that Christians gather on a Sunday.

So Sunday worship begins early. But it's (a) not a day of rest, because most of these believers would need to work, and especially those who are slaves have no liberty about when to observe a Sabbath, and (b) the early church didn't develop a theology that equated Sunday as the 'new Sabbath' and a day of rest.

Sunday did become a day of rest though, in 321. Constantine legislated that it would be a day of rest through an edict, recorded in Codex Justinianus 3, 12, 3. Here it is in the freely available Schaff translation:

On the venerable day of the Sun let the magistrates and people residing in cities rest, and let all workshops be closed. In the country however persons engaged in agriculture may freely and lawfully continue their pursuits because it often happens that another day is not suitable for grain-sowing or vine planting; lest by neglecting the proper moment for such operations the bounty of heaven should be lost

This started the general shift towards Sunday as the day of rest for Christians, but I can't answer definitively when this became associated with a theologising of Sunday as 'the Sabbath', or reinterpreting the Old Testament Sabbath in light of Sunday as rest day. I presume it is a medieval development, and it's clearly the case in Thomas Aquinas. But medieval theology is not my specialty.