If your income is much reduced, would you be entitled to any benefits that may include council tax support?
But yes, you would hope that they would be sympathetic to people who have lost their income due to coronavirus, but that doesn't take away that council tax is a priority bill.
Do make sure you correctly prioritise your income and seek help/keep people informed if necessary. If you have any unsecured debt, it is lower priority than rent/mortgage, food, council tax and other bills, so make sure you cover the essentials first. If you run out of money before you get to any debts, that's the trigger to seek formal help, although if you have a mortgage, look at any payment holidays that your lender may be offering.
Systematic way to go through your budget with signposting to increasing your income, cutting expenses and proritising debts/gettin help:
www.moneysavingexpert.com/family/money-help/
Maybe also look out for Boris's magic financial solution later today, although I would be surprised if there's any meaningful help to most individuals.