This may not work, but check anyway. Go to the following folder on your system. Be aware, the actual path may be altered in this post, but you should be the idea. Or try searching for times.ttf. There are also backups in Windows\winsxs\Backup.
C:\Windows\winsxs\x86_microsoft-windows-f..etype-timesnewroman_31bf3856ad364e35_6.1.7600.16385_none_dd45dd1afaaad8e7
If you find the folder and it is empty, try adding a file called times.ttf. If it is already there or you can add it, try right clicking on it and selecting "restore previous". If this works, it may replace all the Times New Roman files, if not, add the following and do the same.
timesbd.ttf
timesbi.ttf
timesi.ttf
You might also try a subsitute font in Word to see if it will use the Times font.
CG Times (W1)
New York
Times
Times Roman