If a cyclist is on a ‘road’ (that includes a pavement on a public highway) and is riding dangerously or carelessly, they are committing an offence under sections 28 or 29 of the Road Traffic Act 1988, as amended.
The definition of ‘dangerous cycling’ given in section 28 is if the way one rides “falls far below what would be expected of a competent and careful cyclist, and it would be obvious to a competent and careful cyclist that riding in that way would be dangerous”. In this context, “dangerous” refers to danger either of injury to any person or of serious damage to property. What would “be obvious to a competent and careful cyclist” in a particular case includes not only the circumstances of which he could be expected to be aware but also to any circumstances shown to have been within the knowledge of the accused.
The definition of ‘careless and inconsiderate cycling’ given in section 29 is that “if a person rides a cycle on a road without due care and attention, or without reasonable consideration for other persons using the road, he is guilty of an offence”.
http://www.legislation.gov.uk/ukpga/1988/52/part/I/crossheading/cycling-offences-and-cycle-racing
The maximum fine for dangerous cycling is £2,500 and for careless or inconsiderate cycling it is £1,000.