Bath and North East Somerset Council records "35 conservation areas in our district", and on boundaries says "You will often need permission for the demolition of garden walls, especially those facing a highway or public footpath". It adds that "Further controls on Permitted Development rights will apply if we have made an Article 4 Direction", and describes such a direction as applying "in a particular location where the council is protecting special features, such as a conservation area or the setting of a site with high heritage value".
The Planning Portal sets the national rule behind every Bath 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 Bath boundary already exceeds those limits, the condition is that "its height would not be increased".
The same Planning Portal conditions require, at a Bath 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 Bath conservation area it says "you might need permission take down a fence, wall or gate".
In Bath we would explain how you check the conservation boundary, any Article 4 direction and the proposed work before confirming whether permission is needed.