Just had a listen to the archive and I have come down on the side of the controller.
BA got their approval to push conditional on the ryanair passing clear.
2 minutes later the ATR got their approval to push "after the ryanair was on stand" and were cautioned that BA was also pushing back
About 1.5 minutes later there was a clipped tx from ATC advising of "an atr pushing back behind you to point B". That was most likely for BA but the callsign was clipped and nobody responded.
About 30 secs later the BA advised that the ground crew had stopped the push.
It took BA the best part of 4 mins to get moving. That was the problem. The ryanair they should have pushed behind was long gone and had been on the stand long enough for the ATR to push back and move behind the BA. The clipped warning call from ATC probably came about the same time that the BA started to move.
I have no axe to grind with BA, I think their crews are generally excellent but this wasn't their finest moment.