The published recording does not support either side of the argument, either of terms of what was said or the time delay between clearances to push.
From memory of that part of Dublin airport, two aircraft pushed to start points B and C at the same time would have to cross paths, so something should have been said on air so that both crews had full situational awareness. On a busy ground frequency, surely that's the controller's remit?
However, the tone and attitude of the BA pilot was a bit haughty. Who would have thought it...