The 2011 transfer of private sewers moved responsibility for shared drains and lateral drains (the section running from your boundary to the public sewer) to the regional water and sewerage companies. That covers a large proportion of blockages people assume they must pay for.
The practical test is: does this pipe serve only my property, and is it within my boundary? If yes, it is yours. If it serves more than one property, or lies beyond your boundary, call the water company first — Thames Water, Southern Water or South East Water depending on where you are.
If you are a tenant, blockages are normally the landlord's responsibility unless the tenancy agreement says otherwise and the blockage was caused by misuse.