Answer
Lighting a bonfire on private land in itself is not illegal (although it can be if within 30 meters of a road - see SQ625), but the smoke created can be a statutory nuisance.
Lighting a bonfire can be illegal if a person lights a fire, or directs/permits such fire, on land and the fire/smoke injures, interrupts or endangers users of a highway/carriageway.
Otherwise, the smoke created by the fire can be a statutory nuisance, and the environmental health department of your local authority will be able to take action if the smoke is classed as a statutory nuisance. You need to record the details of who is lighting the fires, what time, what the effects were and your details.
If the fires are very irregular then it is unlikely that the council will take any action. The council can stop the person from committing a statutory nuisance and failure to comply can lead to prosecution.
Please see the links in Related Information for further details.