I work for the local council and we have flexible working. For us, as long as you clear it with your manager, it really is very flexible.
So, I work 30 hours a week, same as a colleague who does the same role as me. However, she works basically 9-5, Mon to Thurs, and has Fridays off completely and I work 9-3pm Monday and Tuesday, 9-5pm Wednesday and Friday and 9-1pm on Thursday as that fits round childcare.
As long as it’s reasonable (I.e. I don’t think it would be deemed reasonable to say you want to work all night, and are never available during normal office hours), our place allow it. We are also expected to be flexible in return which is fair enough - so if a meeting has to be booked during a time I’m normally not working, then I will always do my best to accommodate by tweaking my hours if i can.
We can also take up to two flex days a month (so, if we work over our hours, we log it on a spreadsheet and can then take those hours back, either as individual hours, as a half day or as a whole day).
Your council will definitely have a written policy for this which should be in your welcome pack or even available on the internet- if not, email HR and ask for it.