when my cell phone is not connected to the WLAN and has an active connection via mobile data. The app does not automatically update new incoming emails, even though the background synchronization is set to “Always”. If I update manually in the app, the latest e-mails are displayed after a short time.
I haven’t tried it on the new Thunderbird yet, but with K-9 I receive emails just fine when I’m not at home and in my wifi.
So best to post some details like app + Android version, which phone and which internet provider? Maybe your provider plays a role in all this, I really don’t know.