To be fair, in the middle of a battle is not the time to hash out if she had authority over the rest of the fleet. The other ships may have followed her orders as the captains of those ships were wise enough to realize it was not the right time to argue over something like that.
Similarly if the 1st officer of the Defiant was a starfleet officer it would have been inappropriate for the other captains to be like, "I am not listening to you I am a captain and you are only a commander." There would have been chaos in the middle of battle as everyone bickered over who was in charge.
Side note - when I was in the Navy we had a British officer on our sub a couple of times. I have no idea who he was or what he did as I was engineering so we worked in our own little world and he wasn't allowed in the engine room. I always assumed he was just an observer but now I wonder if he ever served as OOD. We always did weird stuff that was never explained to me.