Does migrating to Apple’s new Photos app actually double the storage requirements

Open the Terminal.app and type sudo du -k | awk '$1 > 500000' | sort -nr, this command will search for all Directories bigger than 500MB and sort them by size. So you should find the Location to your Backups.

Photos does not actually move the files or duplicate them when importing from an existing iPhoto or Aperture library. It simply creates hard links to the files in their original location from the new Photos library.

See how Photos saves disk space on Apple's site.

More on hard links from Ars Technica:

A hard link is simply a reference to some data on disk. Think of a file as a combination of a name and a pointer to some data. Deleting a file really means deleting the name portion of that duo. When there are no more names pointing to a particular piece of data disk, then that disk space can be reused."

You can delete your Aperture library without any problems, however you will no longer be able to use Aperture (unless you set up a new library). Any changes made in your existing Aperture library will not appear in Photos, and vice-versa.

Basically the actual image exists somewhere on the disk, and there are two pointers to the image (one from the old library and one from the new library). So long as at least one of those pointers exists, the data is kept. But when the last pointer is removed (i.e. the old iPhoto library itself is removed, then the image is removed from the Photos library), the data is also removed.