Each device, like your Mac or your phone, has it's own Photos Library. Your Mac Library is apparently on an external drive. iCloud doesn't replace the local library-- it syncs the Mac Library with the iCloud Library and the iPhone Library, etc.
It may take awhile for the pictures to fully copy to iCloud-- and the albums seem to be the last things to copy.
When you make a change on any one of these libraries, the change is copied to iCloud and then to the other libraries, so they all stay in sync. If you wish, you can tell your iPhone to only store smaller copies on its own Library, and only copy full sized pictures to the phone when you want to edit or print. You can tell the Mac the same thing, and this saves memory on the devices. (You choose "Optimize Storage" on the iPhone or the Mac.) The full sized images are always stored at iCloud.
If you want to have a smaller Library onboard your Mac, you can create a new empty Photos Library in the Pictures folder on your Mac (close Photos, then option-click the Photos icon) and (in Settings) designate this new Library as your System Library and turn on iCloud with Optimize Storage.) Then wait-- it will fill itself with all the pictures in your iCloud Library, with albums and all.
Keep in mind that iCloud is not a backup service-- if you delete a picture on your phone, it's deleted everywhere. "Recently Deleted" is your only 2nd chance. You still need that hard drive to back up your pictures.
Oops-- I see that I am following FoxFifth.
