The following is from the police –
We are seeing a rise in the use of e-scooters and there can sometimes be a bit of confusion on where they are allowed on the roads and what the rules are. 
Currently it is illegal to ride a privately owned e-scooter on a public road, cycle lane or pavement in the UK. So you can only legally ride the scooter on your own land – e.g. so your garden or driveway.
UK law states that if you are found to be riding a privately owned e-scooter that is not being used as part of a government trial, you could be handed a £300 fixed-penalty notice and, if you have one, six points on your driving licence. We hope this clears some of the confusion up.
More information can be found here
As shown, riding of these privately owned e-scooters are illegal in public areas but this may soon change. The London Mayor Khan is thought to be in favour of allowing these motor vehicles on to our roads. Most of us have had a encountered illegal riders of these machines breaking the law. These machines are a death trap. The riders of these machines are usually youths and criminals who do not comply with the law now and will not do so when they are allowed to ride them in public areas. Anybody riding these vehicles in London’s traffic must have a death wish. Np mechanically driven vehicle should be allowed on our roads without, at least, a provisional driving licence for a motorcycle/car, third party insurance, Road tax and number plates with DVLA number issue. Just imagine these machines dodging in and out of traffic lanes, in heavy traffic. If allowed; riders must follow traffic laws and regulations as to speed is essential. i do not want t see these machines on public roads or in our parks. I predict a lot of accidents involving these gadgets, I may be wrong but from what i have already seen, I doubt it..