Yes, in most of Welling you can extend a house, because there is very little heritage protection to get in the way. Fourteen listed buildings. One conservation area. Hundreds of planning applications a year from people doing exactly what you have in mind. In Welling, the thing most likely to complicate an extension is the clay under the house, not a heritage officer.
Historic England lists 14 buildings in Welling: three at Grade I, two at Grade II* and nine at Grade II. Listing covers the inside and outside of a building, and almost any alteration to a listed home needs listed building consent on top of planning permission, decided with the original fabric in mind. With 14 across a town of about 16,000 homes, the chance that the semi you are viewing is one of them is very small, and the seller's solicitor will know if it is.
Welling has one conservation area, Red House Lane. Inside a conservation area the council takes a closer look at changes visible from the street, some work that would otherwise need no application does need one, and demolition is more tightly controlled. Outside that one area, which covers most of Welling, those extra controls do not apply.
The planning register proves how often people here alter their homes. Bexley Council logged 748 applications for Welling between April 2024 and June 2026, 329 of them in the last twelve months. That is a town of interwar semis being extended one at a time, and whoever does the drawings for yours will have seen your plan many times before.
The ground needs more thought than the paperwork. The British Geological Survey rates shrink-swell clay across Welling at class 3 on its hazard scale, and the whole area reaches that class. Clay of this kind swells when wet and shrinks when dry, so an extension here usually needs deeper foundations than on firmer ground, and the building inspector will want to know where the nearest large trees are. Get a structural engineer's view early. The foundations may cost more than you expect, and the time to find that out is when the budget is drawn up.
In practice, many rear extensions, loft conversions and outbuildings on an ordinary semi can go ahead under permitted development without a full planning application, within national size limits. A listed home, a home in the Red House Lane conservation area or a house where those rights have been taken away will need consent instead. Search Bexley Council's planning register for the house's own history, which shows past permissions, refusals and any condition that has removed permitted development rights. The search costs nothing and takes ten minutes.
Your freedom to extend in Welling is one of its real strengths: the heritage constraints are few, the precedent is everywhere, and the main question is how deep the foundations need to go.