Important note: you will not receive notifications about pending snap updates and the apps will eventually autoclose, potentially losing data depending on the nature of the app. If you want to control when snap updates occur rather than simply disable notifications, see this other answer.

In GNOME you should be able to go to Settings > Notifications > Snapd User Session Agent and turn them off from there.

To manually update snaps you can either occasionally visit Ubuntu Software > Updates or run snap refresh in the terminal. If you are more tech savvy you can also change when the background update checks occur and automate refreshes, though to my knowledge you will still have to close apps before they can be upgraded: https://snapcraft.io/docs/keeping-snaps-up-to-date

Hopefully in the future Snap will evolve to have a more behind-the-scenes update process that only pester the user for rare critical changes, similar to mobile app stores or even Ubuntu's original package manager apt. For now we will have to deal with the degraded user experience snaps have created, or else remove all snaps and rely on apt.