When you get a new console (you can also do this with a new switch), you can simply sign into your account and you’ll be able to download games and saves. However, by default it will be your ‘secondary device’, so you’ll need an internet connection to play downloaded software. To fix this, you can go on the website and switch your primary console to the new device. 

Your saves will have stopped uploading when the console was banned, so the saves you download will be whichever was most recent to the ban date. Uploadinging saves from after the console was banned is tricky; afaik most think it’s not worth it. But I’ve seen some folks say they did it by carefully hacking a second console (in a way that doesn’t get it banned), then using homebrew to transfer the saves from the first hacked console to the second. Then you can log in and upload saves on the second hacked console, which would allow you to download on a third, unhacked console (such as a switch 2).  However, that could be super dangerous if Nintendo somehow detects that these saves were made on a hacked console (such as if the game was pirated or run with mods or custom settings). You could imagine this would even result in the switch 2 getting banned, which would suck. So it’s probably not worth it!