You will be safe in almost any area in daylight - some places I would not want to go at night though...
Unfortunately, I live in the leafy suburbs of West London so nothing near me that fits your bill... Could try some of the street markets in less affluent areas like Brick Lane, which will all be in full swing on a Saturday which means lots of people about for added security & it begins to look pretty grubby when the markets starts closing down at 5-6 pm. Most of the more, erm, "earthy" locations are south of the river and to the east. Once, the river front was a good source as the old warehouses closed down and were abandoned, but that has all been gentrified with luxury housing these days. Getting difficult to find a good ruin these days in London!
The South Bank complex is all brutalist concrete, but as it's the home of a number of cultural enterprises (concert halls, galleries, museums etc) it's not actually a dodgy area - it just looks like one... A few miles west along the river is Battersea Power Station which has been in disrepair for a couple of decades now, but I don't know how close you can get to it (not at all I would expect).
Cheers, Bob.