You can use the Google Drive app on your phone to download a copy of each photo directly to your new phone. They will be in the Downloads folder, and should show up in the Gallery. You can use a file manager app to move all of those pictures to some other folder of your choice (like the "Pictures" folder that is always present on all Android devices).
If you have a lot of pictures, it might be easier to download all of the photos onto your computer en masse, then copy them over via USB connection.