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If you are changing an existing fence to a wall of the same height do you need Building Control approval?
Just thought I would try the portal of all knowledge as google and local council site is not obvious enough for me 
Not sure about building control, but you may well need planning consent. Well worth a phone call to your local office.
I believe it does need planning consent as the neighbour's plot is within a special planning conservation area. I just wondered about building warrant and design of the wall as it is the neighbour that wants to build the wall.
99.99% sure it would be a no. 100% sure in England and Wales.
Is it attached to a house at any point? If not, then no (in England, anyway).
height dependant.
the special planning conservation area might change things.....
Should have done this first...
Reading the guidance it would appear that a wall could require a warrant if its constructed over 1.2m. Schedule 3 - type 17
Whilst England and Wales it doesn't as guidance is given
http://www.planningportal.gov.uk/permission/commonprojects/fenceswallsgates/
Perfect PJM - Sch 3 type 18 sets the rule as you say - it would be over 1.2m so will require a building warrant.
I am buying a house with the missives agreed, the neighbour over the back has approached the seller regarding changing the fence to a wall of the same height. Looks like they have not considered anything as they were talking about starting the work before we took ownership. The seller said his hands were tied as the missives include the fence as viewed so the neighbour discussed building a wall on their ground only. Looks like they probably need planning permission and a building warrant. I have checked the local council portal and they have neither.