Chippenham sits in Wiltshire Council's area, and Wiltshire is the local planning authority a tree notice goes to. On gov.uk the guidance says work to trees in a conservation area requires a section 211 notice to the local planning authority six weeks before certain work is carried out, unless an exception applies.
The same guidance says a tree preservation order prohibits cutting down, topping, lopping, uprooting, wilful damage and wilful destruction without the local planning authority's written consent, and it lists exceptions, among them work on dead trees and branches and on dangerous trees and branches.
The Planning Portal says an outbuilding may qualify as permitted development on an eligible house, subject to limits on purpose, position, coverage and height, with a 2.5 metre overall height limit where it stands within two metres of the property boundary. The same page says listed buildings, designated land and restrictions on permitted development require separate checks, and that the allowances do not apply to flats or maisonettes.
We would explain how you check the conservation boundary, any tree preservation order, any Article 4 direction and the proposed work before confirming whether permission or notice is needed.