The Gmail web interface, Gmail iOS app, and Fastmail web interface all let me search my entire mail archive, almost instantly, and get a chronological list of results I can rely on. I'm interested in self-hosting my mail, but I can't work out how to achieve this.
When I access my Fastmail account over IMAP using the native iOS Mail app, there's a search function. But it's much slower to return results, and it can take 20 seconds or longer for the search results to be "complete" (in the sense that if I have two adjacent emails at time A and time B, I know that no other email satisfying the query between A and B exists). New results intermittently appear in the list as the search is performed.
Notmuch seems like it might solve this problem while I'm on a desktop computer, although I'm not really accustomed to accessing my mail using mutt or emacs. The problem is with the UX of web apps on iOS or Android. I'd prefer a native app with a faster UI and better rich text composition and drafts.
What software can I use to get search performance that matches the most popular mail hosts' webmail apps, if I want to self-host my mail?