Hmm, thanks all for your advice. I did go ahead and try to upgrade one leg, but at the final page of the process the BA site said something like "Sorry, we can't do it, please contact your travel agent."
Having read the UuA section of the "Spending Avios" sticky in detail, I notice there is an exclusion for TA tickets where the TA is based in a different country from your BAEC profile. It looks like my ticket was processed by a CWT office in continental Europe, and my BAEC profile is linked to my home address in London. Does that mean this is a complete no-go? Is CWT able to do UuA's if I ring them up? Or can BA do it if I ring them up? Or will I just have to sit in ET and forget about it...?