There is no one single law. There are numerous Acts of Parliament and Statutory Instruments governing different aspects of law.
The 1989 Road Vehicle Lighting Regulations explicitly include unicycles in the definition of a pedal cycle, which means that the lighting requirements for unicycles are the same as those for bikes.
The legislation that makes it illegal to ride a bike on the pavement (US sidewalk) is the 1835 Highway Act, which actually pre-dates the invention of the bicycle. That Act makes it an offence to drive a carriage upon the footway. Subsequent case law (a court case in the 1870s IIRC, according to an article that I read recently, or it might have been an episode of The Bike Show) has determined that a bicycle was covered by the definition of a carriage given in the Act (the article cited that case as being instrumental in establishing the right of cyclists to use the road).
It’s not at all clear that a unicycle would be covered by the definition of a carriage and I don’t believe there has been any court case to determine this, so AFAICT determining whether or not it’s legal to ride a unicycle on the pavement in England and Wales would require a unicyclist to be prosecuted in a high enough court for the judge to be empowered to make that decision.
IANAL.