How to change Time Machine settings of a partly broken OS (e.g. by manipulating configuration files)
The folders or paths excluded from the Time Machine backup are stored in /Library/Preferences/com.apple.TimeMachine.plist

To include them again modify the file with an editor (TextEditor will not work) or sudo nano /Library/Preferences/com.apple.TimeMachine.plist and remove all entries in the SkipPaths array. In the example below you'd have to remove line 19 to include the (excluded) /Applications folder again.



You then probably have to restart the backup daemon.

If you don't have access to Terminal or an appropriate editor, copy the files and folders manually with the Finder to the backup volume and copy them back after installing the new operating system.