How do I restore my files backed up in Win 10 in Win 8.1?

When I purchase my HP 15 laptop on 10/2/14, it came with Windows 8.1 pre installed. In January 2016 I upgraded to Windows 10. Then, last week, a BIOS file issue crashed my laptop, forcing me to do a factory image recovery process. This solved the BIOS problem, but reverted windows to the factory installed 8.1 version.

I am continuing to encounter a baffling problem that is blocking me from reinstalling windows 10. Apparently, until I successfully reinstall windows 10, I am unable to recover my backed up personal files. Is there any work around solution that would allow me to recover my files from the external hard drive they are stored on whether I am able to recover win 10 or not?