Could Not Set Up Apple Pay - An error occurred while setting up Apple Pay

When trying to set up a new card on Apple Pay, I get the error:

Could Not Set Up Apple Pay:

An error occurred while setting up Apple Pay

This is from the "Add Card" option in the Wallet app on WiFi and mobile data.

When I try to "Add Card" from the Wallet & Apple Pay option in settings, it fails for one of two reasons:

1. When on 4G it says: "Could not connect to Apple Pay, Make sure you're connected to the internet"
2. When on WiFi it says: "Could Not Set Up Apple Pay - An error occurred while setting up Apple Pay"

I had a my cards set up fine on my old iPhone 6 with the same Apple ID. So why, on a new iPhone 8, am I struggling to set things up?