That was actually brought up in one of the behind the scenes specials. Typically the bells would each have individual tones, but in the show they do not. Of course it will continue to move for a short time after ringing, so that would be one way to tell.
And, of course, presumably the family do have a routine.
As we saw by episode 1, Robert was the first downstairs... So presumably the first bell to ring is his. Next were Edith and Mary and then Sybil. Presumably their bells are also in that order; Cora is irrelevant as she'd awaken presumably when Robert does and have breakfast in bed and Violet moved out years ago.
It seems that the only change in routine that there is over all the years is as Matthew moves in, whenever Isobel stays overnight, when George and the other babies grow up, whenever Violet stays, when Henry moves in and Rose's time at Downton though she was presumably informed as to when breakfast time was so would ring around that time too.