Teej you are thinking of S.4 Road Traffic act, unfit through drink or drugs which relates to mechanically propelled vehicles cars, mopeds, lorries etc. A force medical examiner can give evidence that someone is unfit through drink or drugs in the same way a specially trained police officer can.
Pedal cycles are covered by entirely different legislation because they are powered by your legs. It's a good job the OP didn't get in the way of the van with his horse whilst pissed, because that's against the law as well.
It's very rarely prosecuted because most people are just told to get off and push home. The bike, not the horse.