The Twizy has two batteries. the traction Battery - the big one under the drivers seat that is around 60V and powers the motor. The second is a a small 12V battery like from a motorbike. his powers the ligts etc. but if it does not have enough charge it will not trigger the relays and allow the car to 'start'.
The battery is behind the front bumper. Look under the front of the twizy and you should see two bolts to unscrew, Remove these and pull the whole bumper off. The 12V battery is there and as they are used for all the 12v circuits they do not last long. Some people get a year others 5 or more. Apart from the battery terminals there is nothing special about the battery. Just take it to a Motor bike shop and get a replacement. £20- to £40 depending on where you go and the battery manufacture etc. £70 - £120 from Renault and a delay getting one.