Conservation Area legislation
The legislation concerning Conservation Areas come from the Planning (Listed Buildings and Conservation Areas) Act 1990 and there are particular things that apply:
All buildings and their settings (the garden for example) are automatically listed when in a Conservation Area. Therefore, you must make an application for listed buildings consent: this includes structures within the garden.
All trees automatically have Tree Preservation Orders (TPOs). Therefore, you will have to make an application to carry out any works or to fell a tree, unless it is dead or dying and thus causing a health and safety risk
Replacing windows, doors, downpipes or gutters can require full planning consent and it is illegal to put up signage without consent from the Local Planning Authority.
Historic England has useful information for anyone who owns property in a Conservation Area.