I may be wrong, but I doubt that you could teach A-level with only a third class degree in the subject. But anything below that? In fact, when exams were much more rigorous than they are now, those sitting them had frequently been prepared by people without degrees.
Furthermore, the ardently neoliberal, "Thatcherite" Michael Gove should consider who with a First, in particular, he would expect that economic, social, cultural and political system to encourage into teaching. He would have almost as little luck trying to persuade them to become academics.