Road Transport Act 1988
30 Cycling when under influence of drink or drugs
(1) A person who, when riding a cycle on a road or other public place, is unfit to ride through drink or drugs (that is to say, is under the influence of drink or a drug to such an extent as to be incapable of having proper control of the cycle) is guilty of an offence.
(2) In Scotland a constable may arrest without warrant a person committing an offence under this section.
(3) In this section “road” includes a bridleway.
I haven't ever viewed the amendments or rules governing breathaliser etc but always thought it seems pretty ambiguous and potentially unfair (esp when you compare to the highway code).
Edited to say that reading the earlier part of the act, as I suspected the breathaliser refers clearly to motor vehicles. So it would appear to have absolutely no basis in law.