Missing notifications usually come down to a system setting, an account that's switched off, or mail that didn't land in your Inbox. Work through these in order.
Try these, in order
Notifications are off for Newton on this device. On your phone, open the menu, tap the Settings gear, then Notifications. If it says Notifications are turned off, tap it and allow them. On desktop, go to Settings › General and click Send test; if nothing appears, click Open settings and allow Newton.
Focus or Do Not Disturb is on. On iPhone and iPad, check that your current Focus allows Newton. On Mac, Windows and Android, check Do Not Disturb.
The account is switched off. On your phone, go to Settings › Accounts › the account › Notifications. On Mac, go to Settings › Accounts and select the account. Make sure New emails is on. Turning it off on one device turns it off everywhere.
The email didn't arrive in your Inbox. Newton only notifies for Inbox mail. Mail that Tidy Inbox puts in Low Priority, or that a filter or rule moves to another folder, arrives quietly. Check Low Priority, and add senders you want to hear from to Always in Inbox.
Newton isn't running on your computer. Desktop notifications only arrive while Newton is open and signed in. Closing the window is fine, because Newton keeps running in the menu bar or system tray, but quitting stops them. Some company networks and VPNs also block the connection Newton uses for desktop notifications.
You're using Newton on your iPhone or iPad. While Newton is open on screen, new mail shows up in your list instead of as a banner.
Newton isn't getting the email at all. If new mail doesn't appear in Newton either, the problem is syncing, not notifications.
Good to know
There are no notifications for folders or labels other than the Inbox.
Newton for Windows and Linux has no per-account switch. Use the system notification settings there.
Notifications for read receipts, mentions and Recap aren't affected by the New emails switch.