Cycling on a road or other public place (including a bridleway) whilst unfit through drink or drugs carries a fine up to £1000.
There's no breath test for this and no blood alcohol or other legal limits. The key question is simply whether you are under the influence to the extent that you're incapable of having proper control of your bike.
Road Traffic Act 1988, Section 30 states:
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.
In Scotland a constable may arrest without warrant a person committing an offence under this section.