Well, without the answers to some of the questions above I'm going to be guessing a little. But you have one huge advantage. If all your photos are in iCloud, you can easily recover your full library from there:
The easiest way is to create a new empty library on your mac, (in the pictures folder is the normal place). Set this new library as the system library, and make sure iCloud photos is enabled, and your mac will then download everything from iCloud photos library into your newly created library.
First do you have enough space on your mac for your full library to be downloaded again. If so, carry on - if not, take a backup of your current library and then delete it from your mac. ( You still have the previous version on your old Mac, don't you?)
Then:
Hold down the option key while starting photos, and click "create new". Name it Photos Library.photoslibrary, and locate it in your pictures folder. (First screenshot).
Then open the new library - go into photos preferences (general tab), and click "use as system photos library" (second screenshot)
Finally in photos preferences iCloud tab, set it up as the final screenshot. (You need to have enough space on your mac to download all the photos from iCloud)
It will then download everything from iCloud to the new library.