There is a lot of confusion over who is responsible for a blocked drain or sewer. People often think that blocked drains and sewers are the responsibility of the council, but this is not the case. Sewers and lateral drains come under the responsibility of Severn Trent Water Ltd. Private drainage pipes within your property boundary, which only drain your property, are the responsibility of the owner/occupier. See diagram below for clarification.
If you are a tenant, you may want to contact your landlord or managing agent first as unblocking may be covered in your contract. If this is a repeating problem then it is advisable to contact your insurance company (building insurance).
The diagram below explains who to contact when a drain is blocked.