Hi Guido,
So, timeouts are somewhat tricky as they are not execution errors but Google who shuts your execution down - so you cannot easily catch these errors with traditional error handling like try/catch or if statements. See this on how to handle execution errors
https://www.w3schools.com/js/js_errors.asp
I believe it is possible to get remaining execution time, so you could build some logic into your scripts to prevent them from timing out in the first place. And/or use labels to ensure that the script can pick up where it left on next execution if it times out.
A few more ideas I can think of: Have the script do something extra as the very last thing. E.g. ping you. Or, if you only wish notifications when it fails, have it write today's date to a specific cell in a Google spreadsheet. You can then have another script to run in that spreadsheet, that will notify you if the date is not set to today's date.
sigurd