How can I download pictures from an iPhone I don’t have

You can recover your pictures from a backup, of your iPhone, that you may have created on iCloud or on your computer. Did you use iCloud for photos with your old iPhone? OR did you create a backup of your old iPhone when you had it?

Axel F.

Regarding your data, If…

1. You have an iCloud or iTunes latest backup, then there is nothing to worry about --> Restore your iPhone, iPad, or iPod touch from a backup .... If not read more...
2. You have synced your data to iCloud, then what you will get back is what you have synced to iCloud and uploaded to iCloud Drive; like the information that are synced and stored in iCloud are, Contacts, Calendars, Bookmarks, Mail, Notes, Voice Memos, shared photos, iCloud Photos*, Health data, call history, and files you store in iCloud Drive. (*-If you had turned on iCloud Photos, your content is automatically stored in iCloud.)
3. You have imported your Photos & Videos to Mac/PC then they are safely residing there on Mac/PC
4. You have uploaded some of it to Cloud Drives using the Files app (iCloud Drive, Google Drive, OneDrive, DropBox, Flickr, box, AmazonDrive, Adobe Creative Cloud etc). The data remains there.
5. You have not backed up or do not have the latest backup, nor imported nor you have synced nor uploaded your data to cloud drives prior to the situation that you are in, then I am sorry, all your data is lost.

Did you backup your old iPhone? Did you sync photos with iCloud? Assuming you set up your new iPhone with the same Apple ID as you had on your old iPhone, you could either set up the new iPhone from the most recent backup from your old iPhone. At a minimum if you did sync photos with iCloud, again assuming you're using the same Apple ID, you can simply turn on photos in iCloud turning that on and your photos will sync to your new iPhone.

Go to Settings > Tap your name > Tap iCloud > Make sure Photos is turned on

That can only occur if you have a backup of your previous phone which can be “restored” to your new phone.

If you were using iCloud photos this will happen automatically when you sign-in w/ your AppleID

No, you can't unless you have a backup.