Council Tax is legislated to be the 1st of the month, however, I do know from reading on here that some people pay on a later date. The legislation is also for 10 monthly payments, no payments in February and March as that means more time to collect monies owed. The variation to this is moving house and how the bill is recalculated.
If I changed someone to a weekly payment plan I had to enclose a legal letter stating that this was still subject to the usual legislation covering the monthly payment plan because somehow central government didn't think to cover 40 weekly payments.
Due to the Cost of Living crisis lots of people have now switched to 12 monthly payments.
The main thing to bear in mind is this is a government tax collected locally. They can and do take it directly from your income on a fixed amount, no negotiation which does decimate the salary, easy to google attachment of earnings. Legislation also sets out reminder notices and how many you can have, this is not like a utility bill. Do not fall behind. Pay everything you can if you can't pay the whole amount and contact them. They are lovely people who want to help you because their goal is to collect the money within the tax year. I have bent over backward to help people.
Let January go through then ask about changing the date to later in the month.
Edited to say this sounds far more scary than I intended. I just wanted everyone to be aware it should be a priority bill, there is not a lot of wiggle room on this unlike other bills so as workers our hands are tied. It has been a while since I worked in council tax but do look up your local council for things like attachment of earnings. That is only after reminders, final notice, magistrates court for a liability order and then that gives them the power to do the attachment. You want to avoid that at all costs. You can make payment arrangements but think about it like this, if you are struggling to afford council tax now, it always rises every year so you want to try to clear the bill in the same tax year.