Severn Trent supplies the water and runs the sewers here, so it is responsible for the pipes up to your boundary and for any drain shared by two or more homes since 1 October 2011 (Severn Trent Searches). You are responsible for the supply pipe from your boundary to your stopcock, everything inside, and any private drain serving only your home. In a block of flats, shared internal pipework is usually the building owner's or management company's responsibility, so check your lease.
If your blocked drain also serves your neighbours, Severn Trent should clear it free of charge, so we check this first on every Perry Barr drain call. For a supply pipe leak, Severn Trent may give a metered customer an allowance for lost water if the leak is repaired promptly (Severn Trent).