How do i restore my photos from iCloud onto my new phone without restoring from a backup?

There are two ways that photos can be on iCloud. iCloud Photos or iCloud Backup. They will not be in iCloud Backup if you have iCloud Photos turned on. You cannot view the contents of iCloud Backup by using a computer and logging into iCloud.com but you can view iCloud Photos that way. You cannot do a partial restore or a partial setup using an iCloud Backup.

If you had iCloud Photos turned on in your old device or if you still have your old device and can turn it on now (note that it can take a long time to upload), you can then turn on iCloud Photos on your new device and they will automatically sync to it. See: Set up and use iCloud Photos - Apple Support