Protonmail relies solely on Firebase for receiving notifications on Android. While UniversalPush support is probably in the works, it may take some time until users on ROMs without GSF get built-in notifications.

For those that already use ntfy.sh as a push provider for other apps, https://github.com/0ranki/hydroxide-push is a solution to get push notifications of new mail in Inbox.

The service requires a Linux box to run on, and can be deployed as a container or by running the provided binary. Building from source is of course also an option.

The service is a stripped down version of Hydroxide, the FOSS Protonmail Bridge alternative. There are no ports exposed, all communication is outwards. Communications to Proton servers use the Proton API. The service only receives events from Proton servers, and if the event is incoming mail, a notification is sent to a ntfy.sh server and topic of your choice. Other types of events are simply disregarded, and no other processing is done. The sent push event does not contain any detailed information.

EDIT: Starting from version v0.28.8-push7 the daemon supports HTTP basic auth for the push endpoint.

Disclaimer: I'm the author. All of the work is thanks to https://github.com/emersion/hydroxide, I've merely mutilized the great upstream project of most features for a single purpose. Issues, comments and pull requests are welcome!

EDITED: Didn't notice until looking at the image in this post the notification says "ProtoMail" instead of "Protonmail". Changed the image and it's fixed in the latest version, available when the GitHub action finishes