I fear the GDPR regulations (or at least the general interpretation of them) are going to trip you up here.
I would guess, if you've received a letter, the change was made some time ago - and it was on a direct booking through BA (i.e. not a Travel Agent).
If you'd been able to get through before GDPR came in, the checks and access to someone else's personal data may well have been a bit less onerous. Now - well, I can easily see why BA is not unwilling but unable to call you back, because there's a good chance they'll be in breach of the regulations by discussing someone else's booking with you.
Frustrating though this might be, it is the reality of the brave new world we and BA find ourselves in - and, indeed, there are some national companies that are still feeling their way through this and changing the arrangements day-by-day.
As an example, before GDPR arrived I could send my wife into Screwfix (a national distributor of ironmongery) to collect an order placed online as long as she had my card and knew the PIN. Now she's no longer allowed to do that - but she can collect refunds on the same card for returned orders placed prior to the regulations becoming law. At first she was allowed to collect if added to my account as a nominee. Two days later that changed again and she was no longer permitted to do that. It's a complete mess, and it's hardly a surprise that a company such as BA is taking the most conservative approach possible.
I can foresee that you will need a three-way call with your friend and BA also on the line. Writing to Willie Walsh won't change anything about that - you are dealing with the law here, and I can almost guarantee his response will be that IAG airlines always act within it (EC261 claims notwithstanding sometimes!). Given the confusion at the moment, it's difficult to argue against that.