will my photos appear on my new phone if my old one is not backed up

If you

• Were using iCloud Photos on the old iPhone
• Set up the new iPhone to use the same Apple ID
• Turn on iCloud Photos on the new iPhone

then any photos which you have in iCloud Photos should show up on the new iPhone, once it has had enough time to do the synchronization with Apple's servers.

Note: If your old phone was connected to iCloud Photos, and you deleted photos from it, they're gone. A deletion anywhere is taken as a request to delete everywhere.

What to do before you sell, give away, or trade in your iPhone or iPad - Apple Support

If your old phone is not backed up, and is not using iCloud Photos, then you have a few choices. E.g.,

• Enable iCloud Photos on the old iPhone. Give it some time to upload your photos to iCloud. (Use another Mac, iPhone, or iPad connected to iCloud Photos; or the iCould Web interface; to double-check their arrival.)
• Use the Image Capture application on a Mac to copy photos off the old phone.
• Do nothing, and lose all of your photos.

Transfer images in Image Capture on Mac - Apple Support

If you use iCloud Photos and sync between devices, yes. Or, if you transfer directly between devices during set up.