I don't really know why I'm bothering. I don't actually need to use Wi-Fi at home to access the internet on my phone. I have a laptop and a tablet. I just want to do it because you are supposed to be able to.
BT FON is a feature of BT Broadband - so Tilly is still using her broadband, but is using a part that is accessible to any BT broadband customer.
Thus she could go elsewhere and if there was a BT broadband customer nearby she could access that broadband. Access is using her username and password.