If you are Ph.D. in any science subject ,you can become a teacher in any college or university to teach that subject to graduate or post graduate classes. For that B.Ed. and M.Ed. are not required. If you do Ph.D. in education after M.Ed. you can teach B.Ed. and M.Ed. students .For that P.G. in one more subject besides M.Ed. is required. However, these days some institutes are running integrated courses such B.Sc. B.Ed. ,B.A. B.Ed. also. In that case, even a person having qualification as pointed by you, also can be absorbed as that additional qualification will add advantage in your selection for such integrated courses.I hasten to add that requisite qualifications may vary in different countries.
It depends very much on the country and the field you want to teach. In most (if not all) of the European countries you can not teach in elementary, primary and secondary schools without having masters in education. If you teach at universities, the PhD is many times the basic prerequisite to get a job. But if you would like to teach at teachers colleges, you would most likely need the M.Ed. as well. I would definitely check particular country for requirements to get a job you are interested in.