The job description of "Lady's Maid" means you are addressed by your last name only. In this case, "O'Brien." "Mrs." is only used by the upstairs people for the head housekeeper (Mrs. Hughes) and the head cook (Mrs. Patmore) - neither of whom have been married.
There was a scene in which they discussed what Mary calls Anna now, because - technically - she should be called "Bates." But that would create confusion because her husband is referred to as "Bates" because he was Lord Grantham's valet (same level as a lady's maid). And they could NEVER refer to her as "Mrs. Bates" because that would infer she's in a higher level position than O'Brien.
When the servants are all "downstairs," people not normally referred to by their first names ("under" people like James, Daisy, etc.) call each other Mr., Mrs., or Miss). Mr. Carson, Mr. Bates, Mrs. Patmore, Miss O'Brien, etc. I would think that in this case, they need to stop calling Anna "Anna" because of her promotion. But they still do.
Pay attention now . . . there will be a test. (And I obviously need to get a life.)