This is from Kingston upon Thames council tax student discount page.
in further or higher education:
in further education and under the age of 20:
- on a course that lasts three months or more
- attending at least 12 hours a week for supervised study or examination
- where classes are carried out between 8.30am and 5.30pm
- where the course in not carried out by correspondence
So according to their interpretation, you aren't included in any category. Hence, the demand for council tax.
But I am ssuming there are trying to be too strict on the rules, while saying I need to pay council tax during non-course time. I would say get further clarification before paying up. Ask them where it is mentioned that students have to pay council tax during non course time. And ask them as you were a minimum wage earner, what discount I can get ? You might have to pay council tax, but you might be eligible for a discount also.
You will have to pay or you will be prosecuted, so not paying isn't an option and I would advise against it. Council's have the discretion to let go off your council tax, but that might be a tad difficult.
THis might help also. Get in touch with your local Citizen's Advice Bureau and ask what you can do. That is a free service and might be helpful.