Section 30 of the Road Traffic Act 1988, as amended by the Road Traffic Act 1991, provides the offence of cycling on a road or public place whilst under the influence of drink or drugs. It states:
30(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.
The Licencing Act 1872 also makes it an offence to be drunk in charge of a carriage (which includes bicycles), horse, cattle, or steam engine or drunk in possession of a loaded firearm. So don’t do any of those either 😉