Generally, refuse collection is something that Lewisham Council (now a 4 star council, i.e. excellent) gets right.
Problems often occur for houses on corners. The collectors often think the house is in the road round the corner and do not collect the refuse.
Rarely, they cannot collect. I remember seeing one wheelybin nearby completely full of bricks! assuming that the collectors could move it, the handles would just come off the bin if they attached it to the lifting gear!
Keep reporting your non-collection to the Council 020 8314 7171.

I don't think privatisation would help in this case. Many people would just dump their rubbish in the street or on wasteground such as the Shipman Road site used to be.