Ed: And mobility scooters, as Citoyen points out, are definitely powered but treated pretty much as unpowered (as are electric wheelchairs, ride-on mowers and various other machines which move).
No they're not. There are three classes of Invalid Carriages:
Class 1 is a manual wheelchair, handcycle, disability-adapted pedal cycle or similar, and when used by a disabled person has the same rights as a pedestrian.
Class 2 is restricted to 4mph, for use on footways.
Class 3 is road-going, restricted to 8mph, and is required to have lights and (I think) some sort of token registration document. These will have a switch to restrict to 4mph for use on footways. As the law stands, these are forbidden from using cycle infrastructure, which is ridiculous.
There's no accommodation for mobility aids with higher speed limits. A 25kph electric-assist handcycle, for example, is an e-trike in law (and therefore cannot be ridden on the pavement by a disabled person). There's also no allowance for e-assist cycles to be operated by hand throttles for people who can't reliably pedal, other than those grandfathered under the older UK EAPC regulations.
Mowers etc are a special class of motor vehicle, I think.