Have a try from windows.. it is much less fussy about permissions than the Mac.
Getting file permissions fixed on a network drive is tricky and I am not at all sure how Yosemite managed to mess it up..
If you have any applecare at all, ring apple support and let them talk you through it..
Most of what I have suggested to people to fix it are work arounds.. I am just not sure where the problem comes from but Apple might have some better methods.. not that they reveal publically note.. that would be to confess that yosemite is less than perfect.
The work around is fairly easy if you have a USB drive.. same size as all the content on the TC or larger.. do an archive from the airport utility disk tab.. then you can plug the USB drive directly into your computer.. fix the permissions and access your files.. the problem on the TC as a network drive.. fixing permissions is not actually directly accessible to you.