In most cases, other than for farms operating with an Environmental Permitting Regulations (EPR) Permit, nuisance from odour and smoke is dealt with by local authority Environmental Health (EH) Departments.
Under Part III, Section 79, of the Environmental Protection Act 1990, local authorities are responsible for statutory nuisances in their area and investigating complaints. However, if a farm possesses an Environmental Permitting Regulations (EPR) Permit (typically referred to as an IPPC permit), then the Environment Agency is responsible for cases involving nuisance.
Nuisance can include: a state as prejudicial to health arising from, smoke emitted from premises, dust, steam, smell or effluvia arising on industrial, trade or business premises.
Farming: Code of Good Agricultural Practice