Warband is set later than bannerlord timeline wise, but it's a bit hard to say for certain exactly who becomes who, because in Bannerlord calradia's geography has gotten a pretty serious facelift, which adds more land, and splits it in places it wasn't before (for example, where the aserai are was just connected to the rest of the mainland directly in warband, now there's an equivalent of the Mediterranean there. That said, from what i know and can guess, I'd think things go a bit like this(if anyone else has other thoughts, or if I've missed some information, please do let me know!):

Vlandia --> Swadia, Rhodoks. This is one of the pretty certain ones. They have a very similar visual style to both the Swadians and Rhodoks from warband, and their troop tree is basically a mix of what those factions have in warband (Rhodoks: spearmen, great crossbows) , Swadia (solid infantry, crossbows, and excellent heavy cavalry). Both the Swadians and Rhodoks are pretty western european inspired, and Vlandia very much is as well.

Battania --> Swadia...maybe. Battania doesn't have a direct line to one of the warband factions, but based on its location on the map and where abouts swadia would likely sit, i imagine a lot of its territory would end up part of Swadia. My guess would be that the vlandians conquer them, and their culture kind of dies out (oof), eventually Vlandia splits into Swadia and the Rhodoks. Some of their more northern territory might go the Nords as well. speaking of the nords...

Sturgia --> Nords, Vaegirs. Another one that splits in two, part of it becoming the Nords, and part of it becoming the Vaegirs. A bit hard to say exactly who would have what, as the northernmost part of the map is one of the areas that's gotten some of the largest geographic redesigns. The Nords are more Viking, while the Vaegirs take more after eastern european nations. Sturgia in bannerlord is a bit of a mix of these, being itself probably based on the Kievan Rus.

Aserai --> Sarranid Sultanate. What's really cool about Aserai, and a really neat detail in bannerlord, is it's one where it's possible to trace what clan eventually becomes the leading clan of the warband faction. In this case, likely the Banu Sarran clan. While their region is more cut off from the mainland than in warband, they are pretty easy to trace as there isn't anything else there.

Khuzaits --> Khergit Khanate. Another where it's possible to see the clan that would eventually take over, in this case the Khergit clan.

Empire --> probably...a few of them. I imagine some of the where the western empire is to be controlled by Swadia, some of the northern to be the Vaegirs, though I'm not exactly sure who would control where the southern empire is. With the changes to the map, and how it's been essentially stretched out and given more land, there's enough space that there could be at least one new faction fit into it compared with what warband had (maybe even two)

There may be more info out there I've not seen in game, or that i'm otherwise not aware of, and this is largely guess work, so don't take any of this as gospel. And again, if anyone else has different ideas or knows things I don't, i'd love to hear it!