How do I recover all iCloud photos on my new iPhone?

Are you sure that you are signed into the same Apple Account that you were before? What you describe is exactly what you would see if you were inadvertently going to the wrong account. Let's hope it's that simple!

When you had the opportunity, you should have either bought more iCloud Storage or saved the photos somewhere, like to your computers hard drive or another backup service. While we can't know exactly what happened, but running out of storage, then multiple restarts and updates, your photos were clearly deleted. If they're not in the recently deleted folder or in a backup from before the photos disappeared, there is just no where to get them back from.

Yarita6 wrote:

My iCloud storage was full at the time, and my phone went through several updates and restarts.

It's likely that your photos no longer exist.

If you didn't have them stored elsewhere, there is no place to recover them from.