I was concerned to hear about this. People who are appealing a decision about ESA may have to apply for JSA whilst waiting for an appeal to be heard. However, they may not be fit for work whilst doing so. One option is to request an "extended period of sickness" of up to 13 weeks whilst claiming JSA. During this period, you are treated as meeting the jobseeking conditions without actually having to do so.
If you turn up with the sicknote/fitnote however on the day you sign on, you are indicating that you do not meet the qualifying conditions for JSA, and will not qualify for JSA. You have to claim JSA first and then - the next day even - provide a fit note. Showing a fit note the same day is not going to result in JSA being paid. A reasonably sensible decent compassionate JC+ member of staff would explain that to a claimant....
We come across this happening a lot and have had to advise people to get their MPs involved to get JC+ to deal with these claims properly.
Another possibility is to ask for reasonable restrictions to be included in your Jobseeker's agreement/claimant commitment in the light of your condition. So for example, where someone had had a heart attack, had their ESA refused, had been advised by their GP not to work, and was claiming JSA, we would put in writing a list of tasks which were medically inadvisable (heavy lifting, prolonged standing etc) and ask the work coach to respond in writing if they disagreed, explaining the basis on which they were doing so, and copy this letter to the claimant's MP and GP. If the work coach will not change the claimant commitment, ask for it to be referred to a decision maker. If the DM won't change it, you can ask for a mandatory reconsideration and then appeal it to an independent tribunal. In order to avoid a sanction, however, it's important for the claimant to indicate that they will attempt to comply with the jobseeker's agreement until it is changed.
When claiming in this situation, it can be worth asking for a short term advance of benefit - basically, a loan on the benefit you are about to be paid. This means you don't have to wait at least 2 weeks to get any money.
In Scotland, you can apply to the local authority for a crisis grant to tide you over: in England most, but by no means all local authorities, have local welfare assistance schemes which can provide short term grants to help people in this situation.
Finally, social services still have a duty to provide services those people living in their area who may require those services: generally this covers older people, carers, people with children and the sick and disabled. You don't have to be without food or fuel for very long before you are likely to get sick.