Well, I have had BA social media team on the phone, and AA twitter have been very responsive. The upshot is that I think both airlines now think I have the seats I wanted. Won't truly believe it until I get my backside on the seats, but don't think any amount of to-ing and fro-ing with customer services is going to change that.

AA have advised that I check in on their website, using one of the locators (not the one that shows the correct seat assignments for the LHR-CLT leg - it's only for their reference, apparently). They have said that I will get the assignments that are on the other locator - via some combination of pixie dust and unicorn tears, presumably.

The man at BA said I could check in either with the BA app or AA, but I will follow the advice on here and stick with AA.

Thanks all!

Al.