Yes, in most of Wootton you can, and plenty of the neighbours are already doing it. A few dozen buildings are listed. There is one conservation area. Hundreds of planning applications go in every year. The planning office is rarely the bigger problem here. The bigger problem is the ground under the extension.
Historic England lists 30 buildings in Wootton. One is grade I, one is grade II* and 28 are grade II. Listing protects the whole building, inside and out, so almost any change that affects its character needs listed building consent, whatever its size. With 30 listed buildings, very few buyers will end up owning one, but those who do will be dealing with the council over the inside of the house as well as the outside.
The village has one conservation area, named Wootton. Inside it the council takes a closer interest in what changes are visible from the street, and some work that would be routine elsewhere needs permission. Check on Bedford Borough Council's maps whether the house is listed and whether it falls inside the conservation area. The boundary decides which rules apply, and two houses a few doors apart can sit under different ones.
The planning records show 1,168 applications between April 2024 and June 2026, including 541 in the last twelve months. Those numbers show people applying, and say nothing about how many were approved. Even so, 541 applications in a year means a lot of Wootton households want to change their house.
The British Geological Survey rates shrink-swell clay at class 3 across all of Wootton's postcodes. That is a published hazard class for ground that swells when it is wet and shrinks when it is dry. It doesn't rule out building. It means a new extension may need deeper foundations than the house it joins, and a structural engineer should see the plans before the builder quotes. Trees near the footprint make it more important.
For an ordinary house that is not listed and not in the conservation area, a modest extension can often go ahead under permitted development rights, which allow certain work without a full planning application. Larger schemes, and anything on a listed building or inside the conservation area, need consent. Ask Bedford Borough Council whether permitted development rights still apply to the specific house.
Wootton gives you middling freedom to change a house. The planning rules are fairly relaxed away from the old centre, but the clay under every extension still has to be dealt with.