On finnair.com, I can see my booking, with the schedule change and the ability to select another date. (This date is not otherwise available for an award redemption.)
On BA.com's manage my booking, the original flight details are shown. The call centre is aware of the schedule change (16:35->15:50) but says this is not significant enough to qualify for a no-cost change. I don't mind paying for the change, but the date I want is not available.

AY told me I should either talk to BA or ask them to reissue the ticket once I made the change on their website. The question is, will BA do it or will I risk being left with no booking at all.