On the very same website quoted by the op.
https://www.bbc.com/news/uk-48106617
Unless a vehicle is specifically prohibited, it is road legal by default.
I understand it is quite contrary to that after some looking.
There is a permitted list, going back to 1835 which was later amended in 1888 to include bicycles which were by the original law, illegal.
The permitted category of carriage originally meant a literal carriage and the term was stretched to cover bicycles then once more to cover motor vehicles which of course came later.
Legality of un motorised scooters relies on how effectively you can claim them to be covered by the words of the law:
“bicycles, tricycles, velocipedes and other similar machines”
Otherwise not on the list, not legal.
I was looking mostly at this page: https://swiftyscooters.com/blogs/journal/is-riding-a-kick-scooter-on-the-pavement-legal-in-the-uk which points to the legislation.