The responsibility for maintaining underground drains and sewers which are shared by two or more properties lies with the local water authority. In Gravesham this is Southern Water Services Limited.
You should contact them direct for faults within shared drainage.
When drainage serves only one property such as a cesspit or septic tank then the property owner is responsible for maintaining the drainage. This responsibility remains even if the drainage is beneath/on someone else's land or a fault was caused by a third party.
Please note that a block of flats is considered as a single property for this purpose therefore the water authority is not responsible for the drainage of the building. This responsibility lies with the owners of the freehold and the leaseholders/occupiers of the flats.
If you are unsure if the fault is within drainage you are responsible for then you can still contact Southern Water who may send out a team to investigate. Their website also contains information regarding public and private sewers.