As LTS has said above, a cyclist can be charged with being drunk in charge of a bicycle, for which the penalty can be a fine or imprisonment. In the same way, a person can be charged with being drunk in charge of a pram (with child).
Also there are laws relating to bicycles equivalent to the Construction And Use rules relating to motor vehichles. (Now, you didn't know THEY existed, did you?)
The laws for bicycles relate to the carrying and maintenance of brakes, reflectors, lighs, bells or audible warning instruments, and the general safety of the cycle for the user and other road users.
There are also laws relating to the use of mobile phones (as with a car driver, not to be used while the vehicle is in motion) and headphones connected to a music or audio player of any description.

Unfortunately, thje majority of modern coppers are totally unaware of these laws, although most of them have been in force for in excess of 50 years.
Also, if a copper tries to bring a prosecution against little Jimmy for riding his bike with defective brakes, no lights after dark and while wearing headphones connected to his iPod, the whole of the local community would comment that the action would be a waste of time and public money.
However, if, the next day, little Jimmy was to have been killed by some motorist legitimately crossing a junction while Jimmy was crossing against the lights, as mentioned by some posters who seem to think that such action is acceptable, what would be the reaction if no action was taken against the motorist?

If people wish to put themselves in danger by ignoring the laws of the road then that is their privilege. Just don't complain when they are killed or injured.

Similarly, if some cyclist were to hit me while I am walking on the pavement, his feet would not touch the ground until after the Court case against him for as much compensation as I could wring out of him!

Colyn.
Pedestrian, cyclist and motorist, who has passed his HGV1, PSV1 and Police Class 1 tests.