BT will never cold call you about your Internet connection or anthing else to do with your BT account. Nobody other than a scammer will ask to control your computer or ask you to download anything.
The code they send is a genuine code from BT. It is sent to you if you try to change your password.
The scammers know this so if they have your email address, which is probably all over the Internet at various sites you use and your phone number they call you and say they will sent you a code to prove they are from BT.
They then try a reset of your password knowing a code will be sent to you. Once they have done that the then ask you to repeat the code to prove you are the person they sent the code to and if you give it to them they now have access to your BT email account and your BT account.
They now trawel through your email account using it to gain access to your bank etc.
If you did give them them the code you need to inform your bank and any other companies that you use on the Internet and inform them you have been hacked.
You also need to change your email password and check there are no rules or forwarding email addresses set up.
See link
https://www.bt.com/help/email/email-security/how-can-i-tell-if-my-email-account-is-no-longer-secure
