Alright, Fire Emblem Switch has a name, a trailer, and an approximate release period! It's called Fire Emblem Three Houses.
I didn't look too close at the trailer because I'm at work, but it looks a lot like a higher-res Fates. Similar zoom-ins into combat, similar running around a castle (probably a home base?). It looks like there might be some kind of phalanx combat mechanic? There are a few shots at the end that look like a whole group of generic soldiers attacking at once. I may have gotten the wrong impression though since I wasn't paying 100% attention.
Not a TON to work with, but it's far from nothing. I'm pretty excited, waiting warmly for more details.