I've done some research on the motor with only three leads. Isn't the internet wonderful? 20 years ago, the only possibility would have been to make endless phone calls to people who didn't know anything either.
An ordinary electric motor uses a commutator and carbon brushes to switch the current rapidly so that the magnetic field is always torquing the motor. The drawback of that, is that carbon brushes wear out, although in my experience, if the motor is designed sensibly, they're easy to replace. But brushless motors are, I suppose, better. And this is a brushless motor.
The three leads are, of course, for power; it's a three phase motor. To run a three phase motor, you have to keep switching the power on and off in each phase, so that the coils react against the magnets and spin the wheel. But how does the controller know when to do the switching?
A Hall effect sensor, detects the strength of the magnetic field close to the sensor. So it's able to tell the controller what the situation is, and the controller uses this info to control the power to the three power leads. By the way, I think the throttle is also a Hall effect sensor; opening the throttle moves a magnet, and so the sensor can tell the controller to give more power.
But you can also have a sensorless motor, which is obviously what I have here, and that needs a controller that can handle the absence of Hall sensors. I can get that here for £22.79.
So how does the sensorless controller work? It must work by assuming that the wheel is rotating, and that the switching needs to be done at an appropriate rate. Hopefully, if it's clever, it will make that rate slow at first, and then speed up.
First, I'll see what AlienOcean suggest; they've always been good to me in the past.