If you cannot connect to your home network, that would have nothing to do with BT. Their influence should stop at your modem.
Possible testing ideas; I am assuming that your modem and router are two separate devices, since you did not describe your network topology:
1. Disconnect your router from the modem. Test whether you can now connect to your home network (I'm betting you won't be able to).
2. If you are using wireless, test connecting to the local network with an ethernet cable connection. If it then works with the modem disconnected, shutdown your computer and router, reconnect the router to the modem, let it boot, then boot your computer with the cable connection and test the internet.
3. Take your laptop to an internet-enabled public wifi site and test there.
You have to isolate each component as much as possible for testing.