I am still abroad so haven't called them yet. However, I sent them an email and got a response, today. Well, they either didn't read my email properly or don't care about offering a solution, or both. But they were very keen on explaining some basics:

Thank you for writing to the Club about upgrading your upcoming flights.

We have enhanced our website to allow you to upgrade your flight to a higher cabin online, depending on your ticket restrictions.

Each fare type has its own conditions attached, which may include it being restricted, non-upgradeable or non-refundable (however some taxes paid are refundable). Generally the lower the fare the more restrictions there will be.

You can go directly to our online system to manage your booking and review the fares rules of the ticket you purchased, simply enter your booking reference and your last name:

http://www.britishairways.com/travel/mmbhome/

Then click one of your flight to expand the information, then select the 'Upgrade booking' link.

Thanks BA for completely ignoring the whole point of my initial email.

DUS