How can I combine many directories, recursively?

I recently exported a cloud drive to change storage providers. When I exported the drive, I had to download it in 25 GB parts. Directories (especially noticeable with large directories) were split across the parts.

I started with something like this:

/drive/file1
/drive/file2
/drive/file3

And now I have this:

/part-one/drive/file1
/part-two/drive/file2
/part-three/drive/file3

The structure is arbitrarily nested, so there are cases such as these:

/drive/project/archive/file1
/drive/project/file2
/drive/project/archive/settings/extra/file3
/drive/project/archive/settings/extra/file4
/drive/project/archive/settings/file5

Which became this:

/part-one/drive/project/archive/file1
/part-two/drive/project/file2
/part-two/drive/project/archive/settings/extra/file3
/part-two/drive/project/archive/settings/extra/file4
/part-three/drive/project/archive/settings/file5

If this were a path manipulation exercise, the solution would be trimming /part-x from the beginning of each file’s path. How can I restore the original structure of these directories and files?

The files exist on a unix server where I can manipulate them with basic command line tools over ssh. I can dowload the drive to a macOS or Ubuntu machine if I need to.