For a fence/wall/gate, you will need planning permission if:

It is greater than 1 metre in height adjacent to a highway used by motor vehicles. This also applies to any footpath associated with that highway.
In any other case, it is greater than 2 m in height.
If your home is subject to an article 4 direction or to a planning condition.
If your house is listed or within the curtilage of a listed building.
The fence, wall or gate forms a boundary with a neighbouring listed building.

Building Regs

Fences, walls and gates do not require building regulation approval. Whilst building regulation approval may not be required, the structures must be maintained and structurally sound.
In certain circumstances, if the garden wall is classed as a party fence wall and subject to the work you intend to carry out, you will be required to notify the adjoining owner of the work in accordance with the Party Walls Act. This does not apply to wooden fences however.