Oxford City Council says "Oxford has 18 Conservation Areas at present", and adds that "Designation within a conservation area doesn't mean every building will be preserved and that no changes will be allowed, but instead helps to ensure changes respect the area's character and appearance".
Oxfordshire County Council is the highway authority here. Its scaffolding licence page says "You need to apply to us for permission when you need to erect scaffolding, staging, hoarding or fencing to secure a building site", and that you "must apply at least five working days before your start date". The same page says the licence "also includes the use of access machinery, such as cranes and cherry pickers on the highway in relation to building work".
The county's conditions document puts numbers on two of those conditions. At S2 it says "In high footfall areas pedestrian walkways of 1.5 metres must be maintained under the scaffold", and at S4 that "Where pedestrians pass through / under a scaffold, the minimum height of the first lift must not be less than 2.6 metres".
The Health and Safety Executive states that a scaffold is inspected following installation, "at an interval of no more than every 7 days thereafter" and following any circumstances liable to jeopardise the safety of the installation. It says that inspection is carried out by "a competent person whose combination of knowledge, training and experience is appropriate for the type and complexity of the scaffold" (hse.gov.uk).
We would explain how you check the conservation boundary, the county highway licence and the proposed work before confirming what the job needs and who applies for it.