Yes, and plenty of the neighbours already have. Only a few dozen buildings are listed. None of them is in the top grade. Hundreds of planning applications go in every year. The council is rarely the obstacle in Cheam. The clay under the garden is.
Cheam has 48 listed buildings. None is grade I, 7 are grade II* and 41 are grade II. Listing covers the whole building, inside and out, so changing a listed house needs listed building consent on top of any planning permission, and that includes alterations that would otherwise need no permission at all. Across a suburb of interwar semis and terraces, 48 is a small number. If the house you are viewing is one of them, you'll know, because the paperwork comes with it.
There are 2 scheduled monuments in and around Cheam, the strictest level of heritage protection there is. They matter mainly if your land backs onto one.
Between April 2024 and June 2026, 559 planning applications were made in Cheam, 247 of them in the last twelve months. That is a street full of skips and scaffolding. These figures count applications, not approvals or refusals, so they show how often people try to change their homes here and nothing about how often they succeed. Look up the planning history of the house and its immediate neighbours on Sutton Council's planning register. The extension next door is the best preview you'll get of yours.
The ground is where Cheam pushes back. Every postcode is in class 3 for shrink-swell on the British Geological Survey's hazard scale, meaning clay that swells when wet and shrinks when dry. That doesn't stop anyone extending. It does mean foundations need designing for the soil and not guessing at, with nearby trees taken into account, and a structural engineer should see the plans before the builder sees the money. The interwar builders put these houses up on the same ground. A good engineer will want to know how their foundations have coped before adding new ones beside them.
Across most of Cheam's semis and terraces, a rear extension or a loft conversion can often go ahead as permitted development, with no full planning application. Parts of Cheam Village and Cheam Park are in a conservation area, where those rights are narrower and a planning officer will have views on the front of the house. Ask Sutton Council for a lawful development certificate even where you're sure no permission is needed. It is a modest fee for a document a buyer's solicitor will want to see later.
Freedom to extend in Cheam is middling. The planning rules are fairly light, and most of the difficulty is in the clay and the occasional conservation area. You will probably be allowed to build the extension. The foundations are where it will cost more than you expect.