Are you a BT broadband customer??
No, I am not.
You can use BT Wi-Fi as a BT Mobile customer. See the first FAQ here.
Would BTWiFi also include BTWiFi-with-FON?
You can use the SSID. Fon is a separate product which BT provides with it's own service. You might find somewhere in the UK which lacks a BT hotspot but has a Fon one, but the main benefit is abroad. That's hypothetical though - I never found it any use and local, commercial hotspots are a better bet.