New FastMail iOS App Link
It looks like FastMail wrapped their terrific web app as an iOS app but added push notifications. The notifications are almost instant when a message arrives (from my one day of testing).
It doesn’t appear to support multiple accounts or really add anything other than push notifications. I actually really like their web app so this is fine by me. I also don’t use mail alerts, so have no real need for this app.
Also available for Android and watches, I guess.