Basically, I am going to start working (full-time) next week for the first time (I just graduated). My mum, with whom I live with (the council tenancy is on her name) doesn't pay council tax because she looks after my younger brother and is not in work. So the question is this: will I be expected to pay?
You may be expected to contribute, however this will be minimal (at most it will be around 25% of the bill) as you are deemed as a lodger for Council Tax and rent purposes.
Your mum will need to contact your Council and inform them of the change in circumstances. They will probably request payslips (once you have them) to calculate any contribution you may have to make.