The offence is being drunk in charge of a pedal cycle under s.12 Licensing Act 1872 as quoted by pogofish.
"Drunk" is a question of fact for the Court. A police officer would be able to give evidence of opinion on the question as they are legally accepted as "expert witnesses" on drunkeness. The evidence would normally need to be factual - glazed eyes, alcohol of breath, unsteady on feet, slurred speech, unsteady riding etc - and opinion - "They were drunk". The decision would be one of fact, for the Court. There is no objective line in the sand so we can known when we are, and are not, "drunk".
Other cycling offences (such as riding recklessly or without due care and attention) also apply, whether drunk or not although they are "fine only").
Most of the rest is urban myth and bollocks.
There is no requirement for a licence to ride a pedal cycle. Thus you cannot be banned. I guess if you made it a habit, repeated drunk cycling could be "anti-social behaviour" and hence be ASBO'd or similar.
So far as I am aware your driving licence cannot be endorsed for cycling offences (I am aware that there are various stories about coppers threatening to do so. This may be due to (a) ignorance or (b) an attempt at humour or wind-up.)
There is no power to require a breath, blood or urine sample. There is no official "limit" for alcohol in breath, blood or urine when cycling and thus no offence of "cycling over the limit".