Yes, on most streets in Abbey Wood. Heritage protection here is thin. A handful of listed buildings. Two conservation areas. The thing to watch is the clay underneath, and that is a job for an engineer.
Abbey Wood has four listed buildings: none at grade I, one at grade II* and three at grade II. Listing protects the whole building, inside and out, and almost any change needs listed building consent on top of ordinary planning permission. With four listed buildings across the whole area, the chance that the house you are viewing is one of them is very small. Unless you are buying one of the four, listing will rarely affect your plans.
The area has two conservation areas, Crossness and Woolwich Road. Inside them the council can control changes that would otherwise go ahead without permission, such as alterations to frontages, roofs and the look of the street, so extending a house inside one usually means more forms and more scrutiny of the design. Check the council's conservation-area map to see whether the address falls inside either boundary. For most houses it won't.
There is also one scheduled monument in the area. That is the strictest level of protection, and it matters mainly to anyone building on land right next to it.
The neighbours are clearly getting on with it. The council received 526 planning applications in Abbey Wood between April 2024 and June 2026, 265 of them in the twelve months to June 2026. Those figures count applications, not approvals. They show that people here would rather add a room than move.
The real constraint is the ground. The British Geological Survey puts the whole area in class 3 for shrink-swell, a published hazard rating meaning the clay swells when wet and shrinks when dry enough to move foundations. In practice that means deeper foundations for an extension, particularly near trees, and a structural engineer's view before any digging starts. It adds cost to a project far more often than it stops one. Nobody reviews your plans more strictly than the clay will.
Outside the two conservation areas and the four listed buildings, plenty of ordinary rear extensions and loft conversions can go ahead under permitted development, the national rules that let some work proceed without a planning application. Where the size or design goes beyond those rules, or the house is in a conservation area, you will need full planning permission from the council. Ask the council for a lawful development certificate if you are unsure; it is cheaper than finding out halfway through the build.
Freedom to extend in Abbey Wood is fair: the heritage rules are light, but you need to know which side of the two conservation-area boundaries the house is on and what the clay will do.