Terraced and semi-detached properties very often connect into a shared run at the rear. If the blockage is in that shared section, ring the water company before paying anyone.
The exception is damage: if a blockage is traced to something a specific property put down the drain, costs can be recovered from that household.
If you are unsure whether the run is shared, a CCTV survey with sonde tracing establishes the layout definitively — useful evidence in any dispute with neighbours or the water company.