As I was writing this post FastMail released a new app with push notification support, it wasn't really for me as I prefer a unified inbox but if you think it might suit you - check it out.
I recently went through the process of prising my email from the jaws of Gmail and getting it into FastMail, the switch went pretty smoothly but I missed Mailbox's push notifications (even if it gets IMAP support I probably won't use it any more - having my email flowing through another cloud server never felt quite right).
The simplest thing to do would be to set up the email channel in IFTTT, forward all incoming mail to firstname.lastname@example.org then use the iOS notifications channel to push the alerts. The main problem with this is related to privacy; the email body will also, at some point, end up on IFTTT's servers.
To solve this problem I created a custom sieve rule inside FastMail's advanced rules section that strips everything other than the subject. It's pretty simple and looks like this:
This rule should probably go after your junk mail filters (which probably look similar to the below), unless you want to get notified about your junk mail too of course.
I also have emails coming through to a work Google account that I like to get push notifications for (without using the Gmail app). IFTTT is a little limited as it only supports one email address in one email channel so I set up Gmail to forward all my emails to FastMail. I didn't really want all this unrelated email cluttering up my inbox so I added another rule, just after my notification rule, to discard them:
And that's all there is to it really - I get a (nearly instant) push notification that I have a new email and I can open the mail app of my choice when it's convenient. One added benefit seems to be a little extra battery life after turning off fetch & push in the native iOS Mail app.