The best way to get to the high level BT complaints department is to either write directly to the chairman or a member of the board. Or write to the address on the bill. Here is a list of useful BT complaint contact details.
While you are waiting to hear back from these letters go through the normal channels of ringing an advisor and insist on getting the complaint escalated if you are still unhappy with the outcome.
Try and get the BT customer service rep advisors name. However they may only give there first name for security reasons (or alternatively ask for their EIN number). Ask for their managers name and a contact number if they say they are going to ring you back, but most importantly you need a reference number for your complaint.
Once again we take no responsibility for incorrect numbers or information on this page as this info was collected from forums but please comment below if any details are incorrect or you found an alternative way to get through and complain to BT.