Are you using an email client or the Fastmail app? If you are using the Fastmail app, here are some ideas:
In the main iOS Settings screen, scroll way down near the bottom and find the Fastmail settings and open. Be sure that
Background App Refresh and
Cellular Data are both set to
ON if you want to get new emails as soon as possible, no matter where you are. You can also enable
new message notifications on your iPhone/iPad. New messages should then automatically appear from time to time. Also (from Fastmail help):
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iOS: To force a check for new mail, pull down on the message list view.
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You can check how this works by sending yourself an email using the Fastmail app. Be sure to add a short Subject. By default, Fastmail will wait 15 seconds or so after you hit Send to give you a chance to undo sending a new message. You can change this in the Fastmail app “Signatures & Compose > Compose options” screen. When I just ran this test (using a cellular internet connection), I received a new message notification after about 25 seconds, and the email appeared in my Inbox a couple of seconds later.
Bill