In most cases, yes. Radley has very little heritage protection for its size. There are only a handful of listed buildings and no conservation area. The neighbours put in planning applications all the time. You're more likely to be held up by the clay underneath than by the council.
Historic England lists 15 buildings within the Radley area. None is Grade I, three are Grade II* and twelve are Grade II. If you own one of them, you'll need listed building consent from the Vale of White Horse council for most changes inside or out, on top of any ordinary planning permission. Grade II* is the tighter of the two grades here, and changes to those buildings get closer scrutiny. With 15 listed buildings in a village of family houses, nearly everyone is buying a house that isn't listed. Radley also has no conservation area, which is unusual this close to Oxford. You can choose your own front door.
There's one scheduled monument inside the area. That's the strictest protection there is, and it only matters if your plans involve the ground near it.
The planning register shows how busy people are. The Vale of White Horse logged 319 planning applications across the Radley area between April 2024 and June 2026, and 158 of them were in the last twelve months. People here change their homes as a matter of course, and the forms prove it.
The ground needs more attention than the paperwork. The British Geological Survey puts the whole area in class 3 for shrink-swell, which means clay that swells when it's wet and shrinks when it dries. In practice an extension here usually needs deeper foundations than it would on firmer ground, and nearby trees matter because their roots draw water out of the clay. Get a structural engineer to look at the foundation design early, while you can still budget for it. Being free of conservation red tape doesn't help much if the soil has its own rules.
For an unlisted house in Radley, many rear extensions, loft conversions and outbuildings may come under permitted development, which lets you build without a full application as long as you stay within national limits on size and position. Anything bigger, or anything on a listed building, needs consent. Look up the planning history of the specific house on the Vale of White Horse planning portal, because permitted development rights are sometimes removed by a condition on an earlier approval, and the seller may not know that.
Freedom to extend is one of Radley's real strengths. There's very little heritage red tape, and the main things to plan for are the clay and a sensible engineer.