For the benefit of future searchers--I feel your pain. Once you get yourself straightened out, get yourself a *real* disaster recovery backup. File History has been abandoned, sadly.
So anyway, I was forced to give up on trying to restore my File History files normally. I wish I had a better solution for you. The good news is that all the data is there in your backup. The bad news is that every file is date-stamped. It's not a magic bullet, but I used a free little utility called ReNamer at https://www.den4b.com/products/renamer to help.
I wish ReNamer had the capacity to just dump my Data folder into it and let it rip -- depending on how many files you're trying to recover and how organized (or not) they are, maybe you can. But it did make recovery feasible at least, because you can use it to strip out the date-stamps in batches. I found that if you try to do more than three or four thousand files at a time, it starts to struggle. Copy your backup to your new Windows installation and go to work on it.
Good luck!