You can try the software suggested by @oliverpowell and hopefully the data will be saved, but in case it doesn't then contact a data recovery company and see if your information can be retrieved from the drive, and keep in mind that the more you use the drive and try to repair it, the worse the damage and the extraction of files may become.
No offense meant, but this can cost thousands of dollars and isn't entirely true. If the Windows 10 Upgrade unsuccessfully installed, then all that is wrong with the drive is the Windows installation. The drive itself should still be in good health and the information should be retrievable.
In the event that the upgrade process formatted the hard drive (doubtful), there is affordable and even free/trial software you can use to recover information from sectors that Windows may have marked as available to be written to. When you "quick format" a drive, such as is common practice before you install a fresh copy of Windows, all Windows does is marks the entire drive available to be written to. This means your information is STILL on the drive, it is just hidden from you because the impression the OS is getting is that you no longer have a need for that data, thus opening that space so it can write new information there.
https://www.runtime.org/data-recovery-software.htm - NTFS GetDataBack is my program of choice for these instances. I used it for over 5 years at my old job working at a computer/electronics repair shop. It has saved many of my personal customers from losing data they thought they lost due to accidental quick formats or Windows becoming corrupt because an update went horribly wrong (just as in your case).
When I attempted to use the free Windows 10 upgrade back in June it bricked my install of Windows 7 as well. I was able to grab my important data with no problem off my drive by plugging it into another computer via a USB to SATA adapter. The adapters can be bought for about 20 dollars or less. There are more expensive ones such as the "usb docks" that trents mentioned, they are nice if you're constantly needing to plug internal drives into another machine to read them.
I wouldn't contact a data recovery company until it is your last resort, but if the drive is still in good physical health (which it should be since it wasn't dropped or damaged physically) you have nothing to worry about.