I have a Macbook Pro with a broken OS (computer was turned off during an update and won’t boot any more).
The Macbook contains some user data that hasn’t been backed up in a while, including a valuable iTunes library.
Will those data be preserved when reinstalling OS X (Yosemite) from recovery mode, or is there an easy way to get hold of it again?
Reinstalling OS X or macOS from Recovery mode or an Installer will not delete any files.
Yes, it should. The last time I restored an OS (El Capitan) the user data was preserved, but I would definitely recommend booting from recovery then trying to back up before restoring. Once you make the changes, if something breaks then it's too late unless you have a backup. Consider Carbon Copy Cloner or SuperDuper! to facilitate.
As long as you don't explicitly erase then you should be okay. You can also use target disk mode or something else to try and make a quick backup before moving forward.
