Without knowing what kind of electric scooter you were on, I've found the lot for you. Looks like the copper was right:
Electric Stand-On Scooters
These are not legal on the public roads or pavements, they are classed as a motor vehicle. It isn't an offence to draw or propel a vehichle by hand close to the near side or left-hand edge of the carriageway. If the motor is off, the question remains as to whether it is a motor vehicle at all since you do have the option to "scoot" should you so wish. I'm not sure how the law would view this, so please note that you have been warned. As for electric skateboards, you can certainly use them on private land with the owner's permission.
Electric Sit-On Scooters (Mopeds)
This is the first category of EV where the law is clear. Electric Sit-On Scooters are classed as mopeds in UK law and require a registration document, number plate, MOT at 3 years, insurance and a valid tax disc. For our purposes a "moped" is a 2 or 3 wheeled electric vehicle with a maximum design speed of 50 km/h (31 mph) that has lights and indicators. Riders must be at least 16 years old, wear an approved helmet, and have category "P" on their driving licence. If you only have provisional category "P" then you must display "L" plates and have a valid DL-196 certificate of completion of CBT issued by an approved training body.
Electric Motorcycles (Mopeds)
Electric motorcycles are classed as mopeds in UK law and require a registration document, number plate, MOT at 3 years, insurance and a valid tax disc. For our purposes a "moped" is a 2 or 3 wheeled electric vehicle with a maximum design speed of 50 km/h (31 mph) that has lights and indicators. Riders must be at least 16 years old, wear an approved helmet, and have category "P" on their driving licence. If you only have provisional category "P" then you must display "L" plates and have a valid DL-196 certificate of completion of CBT issued by an approved training body.
Light Electric Sit-On Scooters and Motorcycles
Light electric sit-on scooters and motorcycles are classed as light motorcycles in UK law and require a registration document, number plate, MOT at 3 years, insurance and a valid tax disc. For our purposes a "light motorcycle" is a 2 or 3 wheeled electric vehicle with a maximum output of 11kW that has lights and indicators. Riders must be at least 17 years old, wear an approved helmet, and have category "A1" on their driving licence. If you only have provisional category "A1" then you must display "L" plates and have a valid DL-196 certificate of completion of CBT issued by an approved training body.
Standard Electric Sit-On Scooters and Motorcycles
Standard electric sit-on scooters and motorcycles are classed as standard motorcycles in UK law and require a registration document, number plate, MOT at 3 years, insurance and a valid tax disc. For our purposes a "standard motorcycle" is a 2 or 3 wheeled electric vehicle with a maximum output of over 11kW that has lights and indicators. Riders must be at least 17 years old, wear an approved helmet, and have category "A" on their driving licence.