Filing this as per https://bugzilla.mozilla.org/show_bug.cgi?id=1549242#c22
My testing results with Thunderbird Daily 69.0a1 and Beta 68.0b2 to 68.0b3
Update for Thunderbird Daily 69.0a1 on Ubuntu 18.04.2 LTS
I get the doorhanger from the "Green arrow circle" next to the "Switch to the calendar tab" icon and click "Download Update"
I don't get a prompt to restart when the update is downloaded and applied
I have to open Help > About Daily and click the "Restart to update Daily" button
Update for Thunderbird Daily 69.0a1 on Linux Mint 19.1 Cinnamon
Same as above
Update for Thunderbird Beta 68.0b2 to 68.0b3 on Linux Mint 19.1 Cinnamon (Ubuntu already updated using Help > About)
Same as Thunderbird Daily on Ubuntu Linux
I have to open Help > About Thunderbird and click the "Restart to update Thunderbird"
Update for Thunderbird Daily on Windows 10
Only the icon with the arrow inside the green circle appears next to the "Switch to the calendar tab" icon".
I have to open Help > About Daily
Click "Update to 69.0a1 (2019-07-04)"
Update downloaded and applied
Click "Restart to update Daily"
With Firefox the "Download Update" doorhanger drops down from the AppMenu, then after clicking "Download Update" and the update is applied the green circle with the arrow appears letting the user know to restart.
Open the AppMenu and the "Restart to update (Version name)" menu item is at the top.
Click that and Firefox restarts.