I've seen it being said somewhere, although I'm unsure as to where. However, looking at old BL screenshots from 2012, it's extremely 'warbandesque',
The end user look of a game has surprisingly little to do with the engine. If you used the same lighting setup between two engines, it would be almost impossible for even the people who made the engine to tell the difference. When you get lots of games in the same engine that look the same, that's because they've been lazy and haven't changed the default assets at all. A lot of Unreal engine games for example have the exact same motion blur and bloom because they don't know how / can't be bothered to change them from the default settings you get when you create an empty level.
The reason early screenshots look "warbandesque" is much more to do with the fact that many of the background assets are basically warband assets, or look a lot like them. However I can guarantee you that if they had announced a game with a completely different setting, with new assets but in the same engine, nobody would have noticed.
Which one of these looks more like warband to you?
The answer is obviously the first one because the colour scheme is the same and some of the assets look familiar, but other than that there is no way to tell what engine either of these are made in. Engines are mainly a developer thing that generally doesn't impact the end user.
Considering they just finished developing Warband and announced Bannerlord a year after, it's safe to assume that the BL engine didn't exist then, and the the development screenshots were that from the old engine. It takes numerous years to develop your own engine, hence why most developers just use UE4, Unity etc. it's quite complex. You also can't really develop a game until the majority of an engine has been built.
It depends. Someone who knew what they were doing could probably throw together a phong type renderer (like warband) in a few days, since a lot of this stuff is in the public domain. It would be unusable except for showing assets, but that's all they need for making screenshots.
You could be right that they were just using a modified warband as a tool for showing the early screenshots, but I don't buy that they spent 2 years developing the game in this state before realising that warband is horrifying turkish spaghetti that is borderline unusable for modern games.
What
did happen is that in 2014 they announced a switch from the old phong-type shaders to PBR, which is a system basically every game released after 2015 has used, but from what I understand only the engine developer has had to worry about that directly. Most of the assets were retained, and that warband-looking armour in one of the first screenshots is still in the game.
tl;dr I don't think that you can use the excuse that they spent 2 or 3 or however many years in warband before making a new engine, because there is no evidence for it.