Is planning permission required for converting an under stair cupboard into a WC?

No, planning permission is not required, but you may have to notify Building Control where drainage works are carried out.

You do not need planning permission but you do need to inform Building Control ( Building Regs) as you are introducing new plumbing to the property.

it depends on the property if its listed then it will need planing and will defiantly need a building warrant but in any case a phone call to the local authorities will not go wrong

It probably won’t need planning but it might. Main issue is the waste pipes and whether any new pipes will affect your neighbours or the visual quality of your home. You will definitely need sign off by building control - who are normally quite friendly, will know your neighbourhood and your type of house and be able to advise you. They may even visit if you ask - best to ask as you don’t want them telling you to re do things they don’t like As part of your building control application they will often confirm whether planning permission is necessary.