In the 90's I built a couple of electrically assisted recumbent bicycles. One two wheel & the other 3 wheel plus several unusual, if more traditional e-bikes.
I'm sure you've found the 1983 UK EPAC (Electric Pedal Assisted Cycle) regulations:
The Electrically Assisted Pedal Cycles Regulations 1983
And the 2015 Amendment:
Te 2015 amendment opens up more possibilities. It removes the weight limit, does not limit the number of wheels to 3, and although it says it increases the power, 3 wheeled vehicles were allowed up to 250W in the 1983 regs.
The main negative is that it has to be a 'pedelec' - i.e. something that needs to be pedaled in order for it to move. I believe they're not allowed to be controlled by a throttle type control. I don't know if the rules have been amended again, but a lot of 'modern' e-bikes seem to have a 'walk assist' mode controlled by a hand / thumb throttle, up to 4mph.
The 250W is defined as "maximum continuous rated power" and seems to be based on the motor, rather than the controller. When (if) such a vehicle is inspected by the police, they tend to look for the motor rating plate - which ought to say something like "250W S1" (motor duty ratings explained
), even if it can be run at much higher power for short durations.
I have been inspected / queried by the police several times, just because the vehicle looked unusual! Generally, I used motors with a genuine 250W continuous rating - but ran them at 2x or 4x the intended voltage, with a much lower duty cycle. I suspect a 4 wheel electric Kart will get pulled over occasionally - just because it looks unusual!
Another option is to rate it as a Mobility Scooter. For them, the regulation is much less prescriptive, other than an 8mph speed limit on the road. Mobility scooters and powered wheelchairs: the rules
So, you could have a 1,000kW mobility scooter, so long as it's (electrinically) limited to 8mph. They need to be registered to the owner, but there is no approval process nor restriction on who can use one. Surprisingly, you can still drive one even if registered Blind!