Electric scooters are currently classed as ‘powered transporters’ by the government and fall under the same laws and regulations that apply to all motor vehicles.
This means that it’s illegal to use them on pavements, in cycle lanes and in pedestrian-only areas, and it would only be legal to use them on public roads if they could meet the same requirements as motor vehicles (e.g., in terms of insurance, tax, license, registration and vehicle construction), which in practice is virtually impossible.
The formulation of a law specifically covering electric scooters has been under discussion for some time. It looks like there may be some developments in this area in the near future, particularly with the rental trials currently taking place.