The last former school teacher to be Sec of State for Education was Estelle Morris
Last ex-diplomat to be Foreign Sec was Douglas Hurd
We've not had, as far as I can remember, and ex-policemen as Home Secretary, nor any doctors as Health Secretary (in any of the various titles the post has had since WW2).
There is a case for saying (possibly unkindly) that given the Morris and Hurd examples, the very last person you want as a minister is someone who's once worked in the field...