Usually yes, and for most houses the paperwork is routine. The heritage protection here is real but concentrated, about seventy listed buildings across the wider area and two conservation areas. The overwhelming majority of homes are ordinary post-war stock with no special designation at all. Getting on for twelve hundred planning applications went through the surrounding area in the past year, so the system plainly says yes a great deal of the time. The question is not whether you can extend, it is whether you bought one of the few houses where the answer gets complicated.
Listing is where it bites hardest, and it is rare. Of the roughly seventy listed buildings, just one carries the top Grade I rank and four are Grade II star, with the rest, some sixty-five of them, at the ordinary Grade II. If your home is one of them, listing covers the whole building inside and out, and the works most people treat as free, new windows, internal walls, a rear extension, even the wrong paint on the wrong joinery, can need listed-building consent on top of anything else. That is the price of buying the tithe barn or the old farmhouse rather than the semi, and you sign up to it the day you exchange. The vast majority of buyers here never touch a listed building and never think about any of this.
Conservation areas are the wider net, and there are two, at Church Lane in Winterbourne and at Hambrook. Inside one of those the controls tighten a notch even for unlisted homes: some of the permitted-development freedoms you would have elsewhere are trimmed, demolition and certain alterations can need consent, and the look of the street is a material consideration in a way it simply is not on a 1970s cul-de-sac. It is not a ban on change. It is a requirement to make the change in keeping, and to ask first.
In practice that splits the village neatly. Buy an unlisted house outside the two conservation areas, which is most of the stock, and the usual permitted-development rights apply, meaning a lot of extensions and loft conversions proceed without a full application at all, subject to the normal size and siting limits. Step inside a conservation boundary or buy something listed, and the default flips to check-before-you-build. The honest move is to confirm the exact status of the specific address before you draw up plans, because the house next door having done it is not proof that yours can.
Blunt version: this is a middling result for freedom to alter, which is about right. Most houses here you can extend with the ordinary hassle and no more, a few you cannot touch without permission for the smallest thing. Check which one you are buying before the architect starts dreaming, not after.