In most UK universities, the Master of Arts is a free-standing graduate degree awarded by examination. In Oxford (as in Cambridge), the status of Master of Arts is a mark of seniority within the University which may be conferred 21 terms after matriculation. Please note the Oxford MA is about reaching a new status within the University and not an upgrade of your BA or an additional qualification. It therefore has no subject or class, and the nominals 'MA' should be represented in place of, not in addition to, the 'BA' in the holder's signature and on documents such as CVs.
If you have completed a BA or BFA you will be eligible to take an MA in or after the 21st term since you matriculated (i.e. seven years after matriculation). This is not an automatic process and to obtain your MA you must apply to graduate at a degree ceremony (either in person or in absentia). You must have had your BA or BFA conferred in order to take your MA. They can be conferred at the same ceremony provided that sufficient time has elapsed since you matriculated.
You are not at any point eligible to take an MA if your undergraduate degree is an undergraduate Masters degree (MBiochem, MChem, MCompSci, MEng, MMath, MMathCompSci, MMathPhil, MPhys, MPhysPhil).