You are opted in to BTWifi-Fon by default. It is in the Terms and Conditions that you agreed to when you signed up with BT.
To check your status and opt out see these links.
I'm opted out of BT Wi-fi community according to that webpage. Rather confusingly in the lower half of the screen it says:
Opt out of BT Wi-fi - Sorry to hear you're thinking of leaving BT Wi-fi. Remember, you're welcome to join again if you change your mind. To opt-out, enter your details below and click to opt-out:
The Homehub says:
|BT Wi-fi Status:||Not active on your BT Home Hub|
I have to say that what I am experiencing is consistent with me being opted in but the BTFon webpage and the HomeHub contradict that. Is it worth me using the opt out menu just to be sure?
When you search for wireless devices in your home does btwifi or bt-fon come up?
I can see BTWifi-X which is secured and has a weak signal. I can also see BTWifi-with-Fon which is open and an even weaker signal.
From your post that sounds like a neighbours BT WiFi service not yours:
Which part of my post ? the Hub or the website feedback, and what is the implication? Should I use the opt out option on the BT website? I did enter my details when I checked the status.
If you switch your router off and the signals remain, they will obviously be coming from a neighbour.