"Does the PM’s family still pay money for their bills? (eg: water bills)"

Yes, the Prime Minister’s family pay all the normal bills for everyday expenses, such as water bills.

However, the Prime Minister has 2 official residences where they can live with their family:

As these houses are also used for official duties, such as welcoming representatives from other countries, the Prime Minister’s family would not be required to pay to maintain these houses. 

diagram to help explain Does the PM’s family still pay money for their bills? (eg: water bills)