Thanks for those that have responded. The information provided was extremely helpful - here's an update now that I've made progress with this.

After more than a 20 minute wait to speak to someone yesterday and the gentleman being pretty unhelpful (and a bit rude), I didn't fancy just calling up to try again (hence my first post).

I took Hiddy's advice and looked at my booking through MMB on the Finnair website. I managed to add my AA number with no problem at all (it still wouldn't let me book seats saying "the seatmap was unavailable"). I noticed checkmytrip updated almost immediately too (so things were looking promising).

Back on BA, I logged onto MMB and still saw just the BAEC number (although it was shaded blue which I hadn't see before) so thought I'd bite the bullet and call up again. It turns out that my AA number was there (I assume it was behind the scenes somewhere) and the gentleman was able to add seats straight away (at no cost). He couldn't have come across as wanting to be more helpful and polite. Granted it was easier in this case because he was doing what I wanted but, by comparison with yesterday, it does provide a good example of the variability in service that BA seems to known for across the different aspects of its service.

I can now go in and change seats through MMB - so I'm off to find one of the nice long threads about the best seats on the upper deck (I have a free choice apart from 60 A and B)!

Thanks again for the input and help on this.