Cycling on pavements is a big issue for most people. Pedestrians apparently hate people who cycle on the pavement, and cyclists often feel that they have to cycle on the pavement to avoid being killed or seriously injured by motor vehicles on the road.
The law is quite clear: cycling on the pavement beside a road is a criminal offence. However the situation in law owes a great deal to the long history of cycling, and the fact that bicycles appeared on our roads before motor cars did.
Technically it is illegal for children, of any age, to ride a bicycle on the footway beside a road. However Fixed Penalty Notices can only be issued to people aged 16 or over, and it's highly unlikely the police would take a younger child to court for cycling on the footway. In addition, children younger than ten years old are below the age of criminal responsibility, which means they cannot be found guilty in a court of law. But they could still face other punishments, and their parents might also be punished, in severe cases.
If you cycle on footways beside roads with your children, you have no real cause to fear the law so long as you take sensible care and attention, and give pedestrians plenty of space.
The Highway Act 1835 is the original basis for the laws on cycling on the footway of a public highway. Section 72, headed "Penalty on Persons committing Nuisances by riding on Footpaths, &c." originally said:
And be it further enacted, That if any Person shall wilfully ride upon any Footpath or Causeway by the Side of any Road made or set apart for the Use or Accommodation of Foot Passengers ; or shall wilfully lead or drive any Horse, Ass, Sheep, Mule, Swine, or Cattle, or Carriage of any Description, or any Truck or Sledge upon any such Footpath or Causeway ;
...
every Person so offending in any of the Cases aforesaid shall for each and every such Offence forfeit and pay any Sum not exceeding Forty Shillings, over and above the Damages occasioned thereby.
Of course bicycles hadn't been invented in 1835, let alone motor cars. The aim of the law was to make sure that the sides of the highway were kept suitable for pedestrians to use, unlike the usual muddy mess of the main carriageway, churned up by horses, carts, and other animals.
In 1888, bicycles were starting to become seen quite commonly on roads. The Local Government Act 1888 updated the Highway Act to take these new bicycles into account. Section 85 "Regulations for Bicycles, &c." said:
bicycles, tricycles, velocipedes, and other similar machines are hereby declared to be carriages within the meaning of the Highway Acts
which meant that, as from 1888, riding a bicycle on "a footway or causeway by the side of any road" became an offence, the same as if you had ridden a horse or driven a carriage on the footway reserved for pedestrians. This no doubt made a great deal of sense, as bicycles travelled at very similar speeds to horses and carts, and they were wheeled vehicles.
This law remains in place today.
It means that in the eyes of the law a bicycle is no different to a horse-drawn carriage, and so bicycles may only be ridden on the carriagway of a highway, and must not be ridden on the footway. The law also has other effects, for example giving cyclists the same legal rights as horse-drawn carriages to use the public highway (so long as they stick to the carriageway!), and also requiring cyclists to obey road traffic laws defined by the Highway Acts (except those laws that only apply to motor vehicles, like speed limits).
The Road Traffic Offenders Act 1988 introduced Fixed Penalty Notices, on-the-spot fines for road traffic offences, and the Act has a list of the offences that Fixed Penalty Notices can be used for in Schedule 3.
Cycling on the footway was not included as an offence that Fixed Penalty Notices could be used for.
The Act was modified by the The Fixed Penalty Offences Order 1999 on 1 August 1999 to add the offence of "Cycling on the footway" (article 3(3)(b)) to the list of offences that Fixed Penalty Notices can be used for.