It's clear now that what happened to me was that BT's software detected "suspicious activity" which probably consisted of me absent-mindedly logging on to my BT email from 2 different machines (a desktop & a laptop) at the same time, one of which was using a VPN. I suspect that is enough to trip the alert. When I was unable to recall the answer to my secret question I quickly used up my attempts to guess it. That was that. I could tell that full access to my email and specifically the ability reset my secret question had been blocked because the status of email account changed. The status changed from "Active" to "Protected". "Protected" here is a euphemism meaning blocked. Future users who might read this can check if they are in the same boat (unable to reset their secret question) by doing the following. If you can still log in to MyBt do so -
1) Go to https://bt.com/mybt
2) Enter your BT ID and password and log in
3) This will take you to your MyBT homepage
4) Click on "Your products" on the light grey menu bar below the purple menu bar.
5) Scroll down to the section called "Included in your product"
6) Find Email and click the "Manage" button
7) This will open a page showing your email accounts. I only use one.
Next to your email account there will be a long green button. If it contains the word "Active" then you have some other problem. If it contains the word "Protected" then you're blocked and BT doesn't entirely trust your identity. You will have no option but to bite the bullet and ring up their tech support number. When they ring you back they will ask the usually personal details (name, address, etc) and then they'll text you a 4 digit code which you have to verbally report back. This re-establishes your identity. They will then give you a new randomly generated password and return the account status from "Protected" to "Active". You will then be able to log in and reset your secret question.
This problem is now solved. BT rang me back from their Tech Support in India. The technician changed my password from her end. If I now log in to https://bt.com/mybt and do "MyBT>Your profile" I no longer receive the error message. I was able to edit my secret question and enter an answer that I will be able to use if the problem ("suspicious activity on your email account") recurs, which it probably will. Thanks for your suggestions. Problem solved.
Great to see you got this sorted out!