I agree with everything said so far Podmog. Keep going, don't let the school get away with such a ridiculous policy. Definitely write to the Governers, and CC a copy o the Head and the LEA - and insist on a reply.
The school I teach at has no such policy so it is not a Government based one. OK, so we are secondary but I have not heard of it at any of the feeder schools either. Normally, any medication at our primaries are kept by the school nurse for children to get when needed. Teachers don't normally have responsibility - to protect ourselves I guess if anything went wrong. Very sorry, sad thing I know, but some parents will, and do, bring actions at times - NOT suggesting anyone here would, obviously.
Good luck.