Newton gets new Inbox mail from Gmail and Outlook as soon as it arrives, and checks other folders on a schedule. If nothing new is showing up, try these.
Try these, in order
Check now. On your phone, pull down on the email list. On desktop, press ⌘K on Mac, or Ctrl+K on Windows and Linux, and choose Sync emails. You can also choose Check for new mail from the Newton icon in the menu bar (Mac) or system tray (Windows and Linux).
Sign in again if Newton asks.Your session has expired means your Newton sign-in ended: sign in again. On desktop, This account is out of sync, or Authentication Required in Settings › Accounts, means one email account needs reconnecting: click Sign in again or Re-authenticate.
Check automatic checking on desktop. Press ⌘K (Ctrl+K), choose Sync & Data Info, and make sure Auto-check for new emails is on. Set Check interval to anything from every 30 seconds to every 5 minutes.
On iPhone, open Newton after force-quitting it. If you swipe Newton away in the app switcher, iOS stops refreshing it in the background until you open it again. Also check that Background App Refresh is on for Newton in the iPhone Settings app.
Keep Newton running on desktop. Closing the window keeps Newton running and checking for mail. Quitting it stops checking.
Update Newton. Fixes for syncing ship in new versions.
Still stuck? Run a full sync.
Good to know
Instant delivery covers the Inbox. Other folders update at the next check.
A new IMAP account starts getting new Inbox mail instantly about 2 minutes after you add it.
While the app is open, the phone apps check for mail every minute. In the background, iOS lets Newton refresh about every 15 minutes at most.