Wiltshire Council's own published conservation areas layer holds 244 conservation area records for the county, three of which carry the Trowbridge name: Trowbridge (CA/W/202), Trowbridge (Hilperton Road) (CA/W/203) and Trowbridge (Newtown) (CA/W/204). The council's open data description states that "the local planning authority have a statutory duty to designate and manage conservation areas under the Planning (Listed Buildings and Conservation Areas) Act 1990".
The Planning Portal sets the national rule behind every Trowbridge job. It says you will not need to apply for planning permission to erect a new, or alter, maintain, improve or take down an existing fence, wall or gate where it is "next to a highway used by vehicles (or the footpath of such a highway) and it would not exceed one metre in height (from ground level)", or where "it would not exceed two metres in height (from ground level) if elsewhere". Where a Trowbridge boundary already exceeds those limits, the condition is that "its height would not be increased".
Those conditions also require, at a Trowbridge address, that "no part of the site is a listed building or within the curtilage of a listed building" and that "the right to put up or alter fences, walls and gates has not been removed by an article four direction or a planning condition". In a Trowbridge conservation area the Planning Portal says "you might need permission take down a fence, wall or gate".
In Trowbridge we would explain how you check the conservation boundary, any Article 4 direction and the proposed work before confirming whether permission is needed.