This is what Lewisham's site says

Lewisham Council Landmann Road

It's not that helpful but Kensington & Chelsea's site says

Kensington and Chelsea Refuse

So it's not Lewisham's policy but the law. If you are generating a lot of waste then you'll have to pay to have it removed.

Fly-tipping is a problem in the area and it is a criminal offence

Fly tipping

and yes the council and the police should do more about it but that doesn't mean that people who generate significant amounts of waste shouldn't have to bear the responsibility of disposing on it rather than assuming the council (and council tax payer) has to do it for them.