Yes, and there's less heritage in the way here than almost anywhere with a village sign. There are few listed buildings. There are no conservation areas. The neighbours put in planning applications all the time. The clay under the house will have more say over your extension than the conservation officer.
Across the area, 11 buildings are listed. None is Grade I, one is Grade II* and ten are Grade II. Listing covers the whole building, inside and out, so changes to a listed home need listed building consent on top of any planning permission, and the council will care about the windows as much as the walls. That affects the owners of those 11 buildings. The thousands of post-war and later homes around them aren't affected. There are no conservation areas and no scheduled monuments in the area either, so the extra controls that come with those designations don't apply anywhere.
Between April 2024 and June 2026, 1,175 planning applications were recorded for the area, 499 of them in the last twelve months. That's proof that people here alter their homes and keep on altering them. The number doesn't tell you how many were approved, so don't read it as a promise about yours. It does tell you that a neighbour with a new side extension won't be unusual, and neither will you.
The ground needs more thought. The British Geological Survey puts all of Bowerhill's postcodes in class 3 for shrink-swell, its published hazard class for clay that swells when wet and shrinks when dry. On a new extension, that usually means deeper foundations and an early conversation with a structural engineer, especially near trees. It isn't a reason to abandon the plan. It's a reason to put foundations in the budget before the bifold doors. The same survey places the area in class 3 for landslide susceptibility, so give that to the engineer as well.
In practice, with no conservation area overlaying the streets, many modest extensions to houses can go ahead under permitted development, the national rules that allow some work without a full planning application, within set size limits. Anything bigger, and any work to one of the listed buildings, needs consent from Wiltshire Council. Building regulations apply either way. Check the property's own planning history and any conditions on it before you assume the rights apply, because a developer can remove them when the estate is built. Buyers who don't check sometimes find that out after they've hired the builders.
Freedom to extend is one of Bowerhill's real strengths. Your main problems will be the clay and the neighbours, not the heritage rules.