They say they have "a new router with a different name".
This either means they have bought a third party router, which will not transmit the BT wifi signal. Or the BT wifi has been disabled on the account. Or they may have switched providers. Not a lot you can do about it.
Welcome to the BT community forums.
The BT Wi-Fi next door relied on the neighbours having a BT Home Hub router of some description. It sounds like next door has either got their router switched off or have moved to a third-party router. The latter won't offer BT Wi-Fi.
One option might be to use a mobile phone as a wireless hotspot (3G/4G -> wi-fi).
The BT Wi-Fi app will also show access points in your area, which will give you wi-fi somewhere nearby, but not where you are staying.
Please see following link: http://bt.custhelp.com/app/answers/detail/a_id/11387/~/when-will-my-new-bt-hub-be-enabled-as-a-bt-wi...
This should allow the registration for BTWiFi to be recreated and the hub should then start broadcasting the BTWiFi SSID (Network Name)
It could simply be that BTWifi-with-Fon is not currently enabled on the Hub 6 as historically there have been a number of issues with it on the Hub 6.
In the early days it was disabled, then it was enabled but seemed to cause issue with 'standard' WiFi connections and if I recall, disabled again. I no longer use the Hub 6 so can't check what the current situation is.