Given the general trust in American corporations, it's not absurd to think some of the 89% of the population that doesn't use Twitter would be more willing to install an application to follow official communication on Mastodon than to use Twitter.
Anyway warnings are just broadcast on the mobile network here, which works fine.
That can only be described as a loss.
One would likely just subscribe to your city's hashtag for large cities or to your town hall's account for small ones, and that's all. The rest works like Twitter (except there is less risk of important posts getting invisibilised by the feed algorithm).
In my fediverse test, I just subscribed to #Antwerpen (as well as #Antwerp and #Anvers but that's a whole other problem...) and I did get relevant and interesting posts. In their small town my parents are subscribed to the town hall's Facebook account (the only reason they signed up to Facebook) and it would work the same on any other app, except it likely would spy on them a bit less. Mastodon's Android app is not a "more custom" app than Twitter or Facebook's apps are, on the contrary I would argue.
I just can't get used to checking this thing more than once every few days and like Twitter it just doesn't work if you're not constantly reading your feed, because then you're missing most posts.
It's called a website.
It's hard to answer this question without knowing your local city. My town of < 5000 people has the appropriate .gov site set up and maintained; it's one of the responsibilities of the local government. I'd imagine that any local government structure large enough to have evacuation plans in place is also large enough to have a website to communicate those plans on...
The main difference I see though is that since Mastodon provides RSS feeds, I could at least follow the feed of my country's emergency services without needing a specialised app.