I have now had 5 phone calls over a couple of days - each from different numbers. I have added each one to my 1572 blacklist. I too recently had a need to ask for support from BT and in the course of this I was required to tell someone who appeard to be in India all the my details for "security and privacy".
I strongly suspect that these unwanted calls are using information that has been stolen from BT or is being misused by their employees. Unfortunately it is almost impossible to prove this.
I agree that staff ( perhaps overseas) working for BT are misusing that information. BT needs to regulate the employees.
Some of these scam calls can sound convincing when they have information on you but don't trust them. If someone rings up saying they're from BT or any other company for that matter and you feel they are trying to scam you, get personal information, account numbers etc don't give it out.
The staff can see the amount of scam calls happening and telling people to block numbers won't last as the numbers change. If in doubt about any calls, simply tell the person that you believe the call is a scam and that you are going to contact BT on their official number. The agent should be completely fine with this if they are BT staff and although you won't get back through to that agent you were speaking to, you will know the next person is from BT as you have called them on the official number.
Summary: If in doubt, hang up and ring BT on their official number.