If this was ticketed by BA - I.e. you used avios on the BA site - then you contact BA.

you can only change if there are reward seats available on the new flight, and you can check that by just doing a normal reward search via your exec club page. If there aren’t any reward seats available on your new flight then you can’t change and there is no point calling BA. Keep checking tho as that can change. Seeing empty seats on the seat map is irrelevant tho.