BT do not provide or maintain the external network. This is done by Openreach who work for all service providers. BT do not get any special treatment.
As the fault is outside of your premises, you will not be charged.
You cannot cancel without paying the full cancellation charges.
Just report it either by using another phone, or doing it online. BT will not see a fault if its been accidentally connected to the wrong line, so you will have to insist on a visit.
Make sure you ask for a free divert of incoming calls to your mobile, for the duration of the fault, so you will not miss any calls.
Also check on MyBT, to keep a check on any calls that the other person may make, so you can claim a refund.
It should be fixed in three working days. In theory a visit should not be needed, but the Openreach Field Technician may wish to test back from your house, to the distribution point, to check the routing.