Date: 8/6/2020

Right, so I am right - what a surprise!
QS motors - who apparently are fairly well known for their hub motors will sell me an electric motor for about $250 dollars. The motor controller (which I am still not really sure what it does - surely just plug the motor into the batteries and off you go? Add a resistor operated by the throttle if you want to control your speed?) costs more than the motor (somehow!), and postage is about $150. so about $700 for the whole kit (not including the batteries). Unless the batteries and their LED headlight cost £3000, then Im going to tip the police off about an ongoing incident of daylight robbery… 
I think I will plump for a 3Kw motor. this does not sound like a lot, given that my motorbike has 35Kw and is still underpowered. However, electric Kw are different from petrol Kw. The torque is always and instantly there, no waiting around or changing gear for that power. I am fairly sure that 3Kw is all I need as I am only aiming for a top speed of about 30mph. It just needs enough power to not be crushed by a bus when the driver inevitably does not see you on it! Its either that or paint it bright orange - and im considering that! Helpfully Retrospective publish their motor’s stats as a 3Kw motor, and they claim it is of similar power output to that of a 125cc engine. So I feel fairly secure in that. 
The choice to go with a hub motor is also obvious given the choice of bike: the vespa is not chain driven, the engine sits on the swingarm and provides direct power (via the gearbox) to the wheel. Going with a central motor would mean building a whole new drive system for the vespa. I have seen a post by someone who removed the engine and bolted an electric motor where it used to be, which would retain the gearbox and drive chain, but this seems unnecessarily complex.