Depends on the level you want to teach. Primary would mean a BEd (4 years at a university) and secondary would be a BA in Drama then a PGCE (1 year). If you wanted to teach 16+ you can do a teaching certificate part time, but this limits where you can teach. If you wanted to teach at University level you would need a BA and then at least an MA, or preferably a PHD.
Scarlett, do they have specialist drama teachers in primary schools? If so, a degree followed by a primary PGCE is still ok, it doesn't have to be a B.Ed