I would say if you travel most of your way at the Twizy's top speed, ie 50mph, in the winter with adverse conditions, your range of 24 miles can be borderline. If you are prepared to slow down a bit, like doing 40mph, then it's no problems.
Commuting in the winter in a Twizy is no fun. It can be cold and wet even with windows and doors. In the summer it's a different story, although your type of journey, dual carriageway, doesn't show the Twizy in the best light.
If you choose the Twizy for commuting for economic reasons, I would say think again. With the monthly battery rental fee and relatively high insurance premium, an eco box ICE would save you more money, although certainly not so much fun and satisfaction.