I have already set up my new phone, but backed up my old phone to iCloud afterwards. How do I now restore my new phone with my latest iCloud backup?

You will have to erase the new phone and start over again with a new setup.

Settings > General > Transfer or Reset Phone > Erase All Content and Settings

The phone will restart to the "hello" screen. Choose the option to restore from a backup during the new setup.

Restore your iPhone, iPad, or iPod touch from a backup - Apple Support